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Local Models for Strongly Correlated Molecules

by

John Anthony Parkhill II

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Chemistry

in the

GRADUATE DIVISION

of the

UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge:Professor Martin Head-Gordon, Chair

Professor Jhih-Wei ChuProfessor Robert A. Harris

Spring 2010


Copyright 2010by

John Anthony Parkhill II

1

Abstract


by

John Anthony Parkhill IIDoctor of Philosophy in Chemistry

University of California, Berkeley

Professor Martin Head-Gordon, Chair

The most striking and counterintuitive consequences of quantum mechanics play out inthe strong correlations of many-particle systems. The physics of these phenomena areexponentially complicated and often non-local. In chemistry, these strong correlations arevital to even qualitative pictures of chemical bonding, but they grow intractably morenumerous with the number of particles and remain a significant challenge for models ofchemical behavior. Luckily, the strong correlations relevant to most chemical situations canbe significantly simplified and compressed using the heuristics which have been developedby chemists up to present day: ideas like bonding electron pairs, and resonance. In thisthesis we present a convergent and systematically improvable series of approximations to themany-electron Schrodinger equation which exploit these patterns. Two themes dominatethe work: the use of bonding electron pairs as local units for developing efficient models,and an exponential parameterization of the many-electron wave-function.

i

Contents

List of Figures iv

List of Tables vi

1 Introduction 11.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 The Mean Field Approximation . . . . . . . . . . . . . . . . . . . . 41.1.2 The Exact Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Perturbation Theory, and the deficits of Hartree-Fock . . . . . . . . 71.1.4 The Complete Active Space Self-Consistent Field . . . . . . . . . . 81.1.5 The Coupled Cluster Anstaze . . . . . . . . . . . . . . . . . . . . . 101.1.6 Orbital-Optimized CC as an approximation of CASSCF . . . . . . 111.1.7 The Perfect Pairing Model . . . . . . . . . . . . . . . . . . . . . . . 111.1.8 The Elephant in the room: DFT . . . . . . . . . . . . . . . . . . . 13

1.2 Outline of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.1 A Sparse Automation of Many-Fermion Algebra . . . . . . . . . . . 141.2.2 The numerical condition of electron correlation theories when only

active pairs of electrons are spin-unrestricted . . . . . . . . . . . . . 141.2.3 The Perfect Quadruples Model . . . . . . . . . . . . . . . . . . . . 151.2.4 The Perfect Hextuples Model . . . . . . . . . . . . . . . . . . . . . 151.2.5 Dynamical Correlations: The +SD correction . . . . . . . . . . . . 161.2.6 A Density Functional Aside . . . . . . . . . . . . . . . . . . . . . . 16

2 Many Fermion Algebra 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Many-Electron Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.1 Coupled-Cluster Theory . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Tensor Representation and Permutational Symmetry . . . . . . . . . . . . 222.4 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Rank insensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

ii

3 Numerical Stability of Unrestricted Correlation Models 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Modified Equations Which are Well-Conditioned . . . . . . . . . . . . . . . 353.4 Heuristic Understanding of the Problem . . . . . . . . . . . . . . . . . . . 373.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6.1 Singularity in constrained cluster wavefunctions . . . . . . . . . . . 40

4 The Perfect Quadruples Model 444.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 The PQ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4.1 H4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4.2 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4.3 Ethene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4.4 Nitrogen Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 The Perfect Hextuples Model 605.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 The perfect hextuples model . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2.1 Overview of VOO-CC theory . . . . . . . . . . . . . . . . . . . . . 635.2.2 Definition of the model . . . . . . . . . . . . . . . . . . . . . . . . . 645.2.3 Single excitations, orbital-optimization and exactness . . . . . . . . 665.2.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3.1 Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3.2 Cope Rearrangment . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3.3 Bergman Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3.4 F2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76


6 The +SD Correction 786.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3.1 H2O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.3.2 F2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3.3 BeH2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

iii

6.3.4 H4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.3.5 H8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86


7 A Novel Range Separation of Exchange 907.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.2 The terf-attenuated LDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.3 Application to Range Separated Hybrids . . . . . . . . . . . . . . . . . . . 947.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8 Conclusions & Outlook 988.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bibliography 101

iv

List of Figures

2.1 An example of the diagram notation. Each box is a list of 4 integers (linenotation) characterizing the normal-ordered many-fermion operator. . . . . 19

2.2 Number of (symmetrically unique) elements in Tn for 2 ≤ n ≤ 6 for thecalculations performed in Figure 3 and linear least-square fits. The numberof amplitudes are constrained to grow in a cubic fashion for any rank witha three-pair constraint. Note that Dim(T2) = Dim(〈a1b1||b2, a2〉) since aPP-active space is chosen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Wall time consumed in calculating equation 4 for various 2 ≤ n ≤ 6 usingalgorithm 4, and the cubically growing tensors of Figure 2. Horizontal axisis the number of occupied orbitals, and the space has the same number ofvirtuals. The cost exponent for n = 2 and n = 6 are estimated by the slopeof the linear fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1 Smallest Jacobian eigenvalue for various fragment dissociations, in the 6-31Gbasis for all cases except Mo2. . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Lyapunov Exponents along the Dissociation Coordinate. . . . . . . . . . . 363.3 CCD parameters for H2 in STO-3G basis, horizontal axis θ, vertical axis R

(A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Orbital labeling for H2 dissociation. . . . . . . . . . . . . . . . . . . . . . . 393.5 Orbital labeling for ethene dissociation. . . . . . . . . . . . . . . . . . . . . 40

4.1 Correlations in the PQ approximation. Horizontal lines represent differentorbital pair spaces, occupied below and virtual above. . . . . . . . . . . . . 46

4.2 Scaling of methods in the 6-31G basis. . . . . . . . . . . . . . . . . . . . . 504.3 Potential energy curve for the rectangular dissociation of H2—H2 with the

cc-pVDZ basis and restricted 2-pair doubles orbitals. (H-H distance 1 A) . 524.4 Potential energy curve for the symmetric dissociation of water with cc-pVDZ

basis with RPP orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Error decomposed approximately by source for symmetric water dissociation

with RPP orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

v

4.6 Electronic energy of ethene (R(C–H) = 1.07 A) in (12,12) space with cc-pVDZ basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Unrestricted potential energy curve for the dissociation of N2 in the cc-pVDZbasis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1 Scaling of amplitude iteration wall-time with system size. Calculations per-formed on one core of an Apple XServe (Fall 08). Largest system: t-butane(24,24) pictured. Parameters of a linear least-squares fit are inset. . . . . . 70

5.2 Scaling T with system size. Same systems represented by each data point asFigure 1, as included in the supplementary information. Least-Squares fitsby quartic polynomials are inset. . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 D3h→D6h deformation of benzene in the 6-31G basis . . . . . . . . . . . . 725.4 Three representative points along the path of the Bergman reaction followed

in this study. From left to right, Step 0.0 (eneyne), 0.5 (near transitionstructure, and 1.0 (p-benzyne). . . . . . . . . . . . . . . . . . . . . . . . . 74

5.5 Electronic energy along Bergman reaction coordinate. CASSCF(8,8) in the6-31g* basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.1 Simultaneous dissociation of H2O in the DZ basis. Orbitals are those of theGVB-RCC model [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 Dissociation of F2 in the DZ basis, Errors relative to CCSDTQPH (frozen-core). PHSD employs the intermediate-pair approximation. . . . . . . . . . 84

6.3 Insertion of Be into H2. PQSD is built on the (4,4) active space. Orbitalsare those of PQ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4 Graphical depiction of H8 model for α = -0.4, 0.4 Bohr . . . . . . . . . . . 876.5 Automerization of H8. The reference space is (8,8) and basis is DZ. . . . . 88

7.1 Various attenuators plotted for comparison. The first two are equivalent toErf (ω = .3, .4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2 Dissociation of He+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.3 Dissociation of Ar+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

vi

List of Tables

3.1 Linear coupling matrix for the PP and IP exchange amplitudes for ethene(H2C=CH2) at a C-C bond length of 7.50 Awith unrestricted PP orbitals inthe minimal active space in the 6-31G* basis. . . . . . . . . . . . . . . . . 39

3.2 Linear coupling matrix for the PP and IP exchange amplitudes for fluo-roethene (HFC=CH2) at a C-C bond length of 7.50 Awith unrestricted PPorbitals in the minimal active space. . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Linear coupling matrix for the PP and IP exchange amplitudes for nitrogen(N2) at a N-N bond length of 7.50 Awith unrestricted PP orbitals in theminimal active space in the 6-31G* basis. . . . . . . . . . . . . . . . . . . . 42

4.1 CASSCF energies for symmetric water dissociation (Eh), and relative errorsof PQ, and non-local active-space models with the RPP orbitals. . . . . . . 53

4.2 Correlation Energies for symmetric water dissociation (Eh) with the RPPorbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Correlation energies for dissociation of ethene(Eh) with restricted PP orbitals. 544.4 N2 Total energies (a.u.) and the error relative to CASSCF. (* includes

singles) with unrestricted PP orbitals. . . . . . . . . . . . . . . . . . . . . . 58

5.1 Energetic effects of the hybrid gradient with several OO-CC models of waterwith the 6-31g** basis. Energies reported are Model - CASSCF(8,8) in Eh.R(O-H) = 1A, and ∠(HOH) = 103.1. . . . . . . . . . . . . . . . . . . . . 67

5.2 Typical energetic effects of the intermediate-pair approximation. Frozen PP-orbitals and 6-31g* basis employed in all cases. . . . . . . . . . . . . . . . . 69

5.3 Energies along the D2h coordinate of the Cope rearrangement in the (6,6)space and 6-31g* basis. Total energies are given for CASSCF in Eh, whiledeviations from CASSCF in Eh for PP, PQ, PH. . . . . . . . . . . . . . . 73

5.4 Study of Bergman Reaction in the (8,8) space 6-31g* basis. Total energiesare given for CASSCF in Eh, while deviations from CASSCF in Eh for PP,PQ, PH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 F2 dissociation in the DZ basis and (14,14) space. Total energies are givenfor CASSCF in Eh, while deviations from CASSCF in Eh for PP, PQ, PH. 76

vii

6.1 The H4 model system. Zero is a 2-Bohr square of hydrogen atoms, and .5 isthe linear configuration. The orbitals are those of PQ. The reference spaceis (4,4) and basis DZP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2 Automerization of H8. The basis is DZ. MRPH employs the intermediatepair approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.1 Mean absolute error (kcal/mol) of G2 set atomization energies and errorsof dimer cation asymptotes for various functionals. *(ωB97X), **(ωB97)†(pure Becke 88 exchange [2] and LYP [3] correlation, errors in this row areupper bounds.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

viii

Acknowledgments

Many other people have been an vital part of this work. My advisor took a chance onme with little background, was supremely supportive and I’m deeply grateful. The facultyat University of Chicago were also quite inspirational during my undergraduate studies.I enjoyed the assistance of generations of fellow students (Alex Sodt, Ryan Steele, KeithLawler, Greg Beran, and Daniel Lambrecht). I would also like to thank the professors whogenerously contributed their time to sit on this committee.

This work was fiscally supported by the Department of Energy through a grant underthe program for Scientific Discovery through Advanced Computing (SciDAC). This workwas supported by the Director, Office of Science, Office of Basic Energy Sciences, of theU.S. Department of Energy under Contract No. DE-AC02-05CH11231.

1

Chapter 1

Introduction

1.1 Context

”What I cannot create, I do not understand” - Richard Feynman

Near the turn of the last century it became clear that an adequate description of matter atsmall scales required [4–9] generalizations of the classical notions of ”state” and ”observ-able”. The resulting quantum theory [10–18] revolutionized science, and barring relativityprovides a comprehensive theory of all chemical phenomena in principle. Realistically,the extent of insight that can be afforded from these first-principles depends on how effi-ciently and faithfully the calculation of a molecular wave-function can be performed by thewould-be theoretician. The goal of our work is to make that calculation more accurate andaffordable, for the special case of electrons in molecules.

The mathematical structure of quantum mechanics postulates that observables corre-spond to the action of linear operators on a Hilbert space, a certain sort of inner-productvector space. The state of the physical system is postulated to correspond to a vectorin this space and the corresponding dual vector guaranteed by the Riesz representationtheorem. These are colloquially called a ket and bra respectively, and together a wave-function. The vectors themselves are not measurable. Concerning ourselves mostly withtime-independent problems we will usually imagine without explicitly noting that thiswave-function is prepared as a position-space vector. In this case the bra and ket areboth complex-valued, square-integrable functions of position, |ket〉 ∼ |Ψ(r1, r2, . . . , rN)〉and 〈bra| ∼ 〈Ψ∗(r1, r2, . . . , rN)|. The probability of observing a particle in at a given po-sition is then given (up to normalization) by the N-1 particle integral over all but onecoordinates ρ(r) =

∫dr1...drN−1|Ψ|2.

To begin, we assume that all nuclei have negligible volume and infinite mass (theBorn-Oppenheimer approximation) and their contribution to molecular energies is simpleCoulombic repulsion. Even these basic approximations are too coarse for some interestingchemical problems. We solve the Schrodinger equation (SE, Eq. 1.1) for a many electron

1.1. CONTEXT 2

wave-function |Ψ(r1, r2..., rN)〉 given a electronic Hamiltonian parameterized by the nucleiand the absence of any other fields. We choose the atomic units and they will be omittedthroughout.

Helec|Ψelec〉 = Eelec|Ψelec〉Helec = −

∑i

1

2∇2i −

∑i

∑A

ZAriA

+∑i

∑j>i

1

rij. (1.1)

The first term describes the kinetic energy of the electron, the second their attraction tothe nucleus and the third their repulsion from each other. Exact solutions to this equationhave a complex analytic structure, with derivative singularities anywhere the positions oftwo electrons are the same and are not generally available.

To make progress we must guess state-vectors which we can easily express. To thisend we introduce an orthonormal basis of single electron states χi which we assume wecan grow until until complete. Ignoring relativity, we postulate that each of these states isthe product χi(r, σ) of a spatial function and a spin function which takes on one of twopossible orthonormal vectors with eigenvalues (±1

2). The spatial functions can be expanded

with any sort of square-integrable functions we choose. The simplest wave-function ansatzis a simple Hartree [19–22] product of these 1-electron functions:

|Ψ(r1, r2..., rN)〉 ≈∏i

χi(ri) (1.2)

However Fermions have an eigenvalue of -1 under parity (permutation of coordinates) andthis trial function does not have those statistics. The simplest possible modification whichensures anti-symmetry is a linear combination of these Hartree products, summed over thepermutations σi on electrons with a sign factor for each index swap in that permutation.

|Ψ(r1, r2..., rN)〉 ≈∑σk

Sign(σk)∏i

χi(rσk(i)) (1.3)

This model is called a Slater [23–25] determinant, and it has the important property thatat any given position an electron excludes others of the same spin.

There is an alternative notation for an anti-symmetric product of single-particle states interms of creation a†i and annihilation ai field operators, called ”Second Quantization”[26, 27]. In this notation, which we will rely upon heavily for the remainder of this work,the physical state coordinates ri are abstracted away and replaced with summed indicesover single-particle basis states. In this thesis these single particle states are imaginedto be vectors in the position representation; for this reason we will omit resolutions overposition space. The representation chosen for the single-particle states (position eigenstates,momentum eigenstates, etc.) usually doesn’t alter the physicality or notation of a state insecond quantization.

1.1. CONTEXT 3

The creation and annihilation operators are not a part of our intuitive reality (ie: theyare not Hermitian, and don’t possess real spectra), but so long as they span a space whichsatisfies the hypotheses of our Hilbert space and obey the canonical commutation relations(we construct them to do so), they provide an allowable representation of our quantummechanical system. We call the space they generate by action on the vacuum a Fockspace. The Fermion statistics are book-kept with the anti-commutation relations, andnormalization conditions which define the algebra on these operators:

Commutation: [ai, aj] = [a†i , a†j] = δi,j (1.4)

Anti-Commutation: [ai, aj]+ = [a†i , a†j]+ = 0 ; [ai, a

†j]+ = δi,j (1.5)

Normalization: aiai = a†ia†i = 0; aia

†i = 1 (1.6)

The eigenstates of the annihilation operators are those states which are unchanged by de-tection, so-called coherent states but this thesis is not concerned with their properties.Second-quantization makes the expression of many-particle states significantly easier be-cause we don’t need to constantly rely on the complex (3N variable) analytic form of thewave-function to express the operator algebra which is the essence of our quantum physicalsystem.

To employ this notation we assume the existence of some vacuum |0〉, a state of zeroparticles. The Slater determinant is written as a simple product of creation operators onthe left of this state; each populating a previously vacant single particle state. It is usefulto introduce some conventions on the indices we will be using. Any sort of orbital may bedenoted: p, q, r..., occupied orbitals are denoted: i, j, k..., and virtuals a, b, c.... Theatomic orbital basis will be denoted: µ, ν, λ.... Any observable N-particle operator canbe written as a tensor of dimension 2N in second quantization. Commonly we will write anoperator in that fashion with a Xket

bra ie: 〈asar|Xrspqa†sa†rapaq|a†pa†q〉 denoted: Xrs

pq . The Slaterdeterminant can be denoted:∑

σk

Sign(σk)∏i

χi(rσk(i)) =∏i

a†i |0〉 (1.7)

This approximate form for the many-electron wave-function provides a useful starting pointfor quantitative calculations and so we should describe how it may be calculated in detail.There is a special class of operator strings, called normal-ordered, which vanish when theirexpectation value is taken with the vacuum. In a normal-ordered operator string all creationoperators lie to the left of all annihilation operators. Although the utility of this class ofoperator strings isn’t yet obvious it will be explored at length in Chapter 1.

1.1. CONTEXT 4

1.1.1 The Mean Field Approximation

Starting with a model of 1 determinant for molecular electronic structure, we hope toapproximately solve the SE. In most calculations the single particle basis is written as alinear combination of atomic orbitals (φµ(r)) which are themselves expanded as sums ofGaussian functions.

χi(r) =∑µ

Cµi φµ(r) (1.8)

The atomic orbitals are non-orthogonal with overlap matrix Sνµ = 〈φµ|φν〉. The first twoterms of the Hamiltonian, Eq. 1.1, do not present much difficulty so long as the right basisfunctions are chosen they are simply integrated directly. We denote the matrix elementfor the ”core” Hamiltonian (kinetic and electron-nuclear terms) hji = 〈χi| −

∑i

12∇2i −∑

i

∑AZA

riA|χj〉. The mean-field approximation is equivalent to supposing that each electron

experiences an effective 1-particle (2-index) Hamiltonian parameterized by the others. Theinter-electronic repulsion term has second quantized representation: V rs

pq = 〈sr|1r|pq〉. There

are two two-index contributions:

J ji = 〈ij|1/rij|ij〉 =

∫dr1χi(r1)[

∫dr2|χj(r2)|21/rij]χi(r1) (1.9)

Kji = 〈ij|1/rij|ji〉 =

∫dr1χi(r1)[

∫dr2χj(r2)1/rijχi(r2)]χj(r1). (1.10)

So we construct the best possible effective 1-particle Hamiltonian, the Fock operator:

f ji = hji + J ji − Kji (1.11)

It should be immediately obvious that the mean-field wave-function is not an exact solutionof the many-electron problem, because it doesn’t even depend on the complete many-electron Hamiltonian. However in the next section we will cover the exact solution. Thedeterminant which is the eigenfunction of this mean-field hamiltonian can be determined[28–30] by solving the generalized eigenvalue problem:

fC = SCε (1.12)

ε is a vector containing the eigenvalues of f corresponding to each orbital. In the sim-plest Roothan-Hall [31, 32] approach the generalized eigenvalue problem is simply solvediteratively. Each new Fock operator generates a new determinant which generates a newFock operator until convergence. At each iteration the N lowest eigenvalue orbitals arechosen to be occupied, and the remainder are called virtual. The Hartree-Fock energy is

1.1. CONTEXT 5

the expectation value of the true Hamiltonian with the converged determinant:

EHF =∑i

εi +1

2

∑i,j

J ji −KJi (1.13)

Orbital Rotations

One can arrive at any determinant of orthogonal spin-orbitals in a given basis from anyother by a series of unitary transformations of those basis vectors. Any unitary transfor-mation can be parameterized as the exponential of an anti-Hermitian operator. In this waywe can express the converged HF orbitals in terms of an orthogonal set of guess orbitalsas:

C = C0U = C0eθ (1.14)

δEHFδU i

p

= 2f ip (1.15)

θ = −θ† (1.16)

the (NMO, NMO) matrix θ can be geometrically termed a rotation operator which mixestwo single-particle states. Using this orbital rotation picture we can develop a steepestdescent algorithm to solve the HF equations, by following the gradient of the energy withrespect to θ. At each iteration the chain rule is used to construct δE

δθfrom δE

δUand δU

δθ. The

Hartree-Fock energy is invariant to rotations within the occupied and virtual spaces, butnot between them.

1.1.2 The Exact Solution

The mean-field solution provides an orthogonal 1-particle basis partitioned into occupiedand virtual spaces, and an approximate wavefunction which lies entirely in the occupiedspace. Within the given 1-particle basis the space of N-electron wave-functions is spannedby a linear combination of all possible N-orbital determinants. These can be imagined asresulting from the action of normal-ordered ”excitation operators” which replace a certainnumber of indices, occupied in the HF wave-function with previously unoccupied 1-electronstates.

|Ψ〉Full-CI =∑

p1,p2,...pN

Cp1,p2,...pN

∏i

a†pi=∑n=1..N

Cn · |ΨHF 〉 (1.17)

Where: Cn =∑ik,bk

Cb1,b2...bni1,i2...in

a†bn ...a†b1ai1 ...ain (1.18)

1.1. CONTEXT 6

Given this ansatz of all possible configurations, the Full-Configuration Interaction ex-pansion (FCI) [33–40], the ground state wave-function can be determined via the Ritz vari-ational principle. In the simplest possible version the Hamiltonian is explicitly constructedand diagonalized in the basis of Cn providing a complete manifold of bound states.

H =

EHF 〈0|H|C10〉 · · · 〈0|H|Cn0〉

0 C∗1C1〈µ1|H|µ1〉 · · · C∗1Cn〈µ1|H|µn0〉...

. . ....

0 C∗nC1〈µn|H|µ1〉 · · · C∗nCn〈µn|H|µn0〉

(1.19)

Where: µn =∑ik,bk

a†bn ...a†b1ai1 ...ain (1.20)

If only a few lowest-lying states are desired the matrix can be implicitly diagonalizedusing a Lanzcos-type [41, 42] algorithm and only the lowest vector must be stored. How-ever each vector must be represented in the basis of all possible n-electron determinants,a space which grows as NMO!. To be concrete, imagine a water molecule with 8 valenceelectrons. Minimally we may introduce two single-particle basis functions for each electron,although chemical accuracy will not be achieved until much more than 30 are added. Evenin this insufficient basis there are 16!

8!(16!−8!)= 12870, 8-electron determinants. If fifteen basis

functions are introduced there are roughly 782 gigabytes worth of determinants in the fullexpansion. If thirty basis functions are introduced for each electron there are ≈ 2.4 ∗ 1014

expansion coefficients and so on. It is unlikely that more than 18 electrons can ever betreated in this fashion within the author’s lifetime even with a basis which is very small.

The difference between the HF energy and the FCI energy defines the correlation en-ergy. The remaining problem of this work is to find truncations of the FCI expansion whichcan be expressed in a number of variables which can be practically manipulated on a mod-ern computer. A multitude of possible approximations have been introduced, many withsignificant success and application to chemistry. The variational theorem ensures that FCIenergy provides a lower bound for any approximate model which is determined by projec-tion against the whole SE, and so often the cheaper approximate models will underestimatethe correlation energy. We will review some established approaches before proceeding intoour own work.

Size-Consistency

The most obvious first step to a tractable approximation of the electronic structureproblem is limiting the rank of Cn and solving the SE over that incomplete space. Thehighest-rank excitation operators are the most costly to calculate, O(m2n+2), and often lesssignificant than the less numerous, lower-rank contributions. Unfortunately the accuracy of

1.1. CONTEXT 7

such a truncation varies when two non-interacting systems are treated as a unit rather thanapart. In chemistry, where particles are redistributed during the course of reactions betweenreactants and products (which we can imagine to be independent), approximations whichare not size-consistent lack a significant cancellation of errors and are simply less-useful.

Intermediate Normalization

Often the Hartree-Fock wave-function has significant weight in the exact expansion ofthe wave-function, and there is a convenient choice of normalization, Intermediate Normal-ization, reflecting this fact which sets the weight of determinantal reference to 1. Formally〈Ψ|0〉 = 1 where Ψ is some correlated improvement, and the energy of concern (now thecorrelation energy) is given: 〈Ψ|H|0〉 = Ec. Throughout the remainder of this work welargely imply this choice of normalization, and if not otherwise noted assume every energyis a correlation energy.

1.1.3 Perturbation Theory, and the deficits of Hartree-Fock

The true many electron state includes information about electrons excluding each otherlocally, and spin-coupling information which cannot be represented in a single determi-nant or effective 1-particle field [43]. Conversely the exact answer has all of these effects,but defies expression and so we seek something simpler. The simplest way to correct forcorrelation is to perturb the HF wave-function with the missing components of H and cal-culate the leading order perturbative correction [44, 45]. H|ΨHF 〉 is spanned by the spaceof double excitations above the reference, because H is a two-body operator. As in everyperturbation theory we introduce a partitioning of the Hamiltonian: H = H0 + V , in thiscase (the Møller-Plesset partitioning) we choose H0 = f . In kind we partition the space ofmany electron determinants into space (0) which is spanned by the eigenfunctions of H0

such that E0 = 〈0|H0|0〉 and the remainder. Given these choices the correlation energyand corrected wave-function are determined as usual in Rayleigh-Schrodinger PerturbationTheory (RSPT) by collecting terms of each given order from expansion over the SE.

(H0 + V )∑k

|Ψk〉 = (∑k

Ek)∑k

|Ψk〉 (1.21)

1.1. CONTEXT 8

We introduce the projectors onto each space: |0〉〈0| and P = 1− |0〉〈0|. We determine theamplitudes of the first order interacting space by projection onto the perturbation:

|Ψ〉MP1 = (1 + T2)|0〉 (1.22)

= −P (H0 − E0)P V |0 > (1.23)

→ T2 =〈0|V a†ba†aaiaj|0〉εa + εb − εi − εj

(1.24)

Then calculate the energy of the corrected wave-function with the SE:

Ec,MP2 = 〈0|V |ΨMP1〉 = 〈0|V T2|0〉 (1.25)

Strong Correlations

The MP2 wave-function significantly improves on HF, and gives nearly quantitativemolecular interaction and atomization energies. However there is serious cause for concernin the details. In cases where the MP2 correction is large, one might be tempted tocontinue the series to higher order to achieve greater accuracy, but such an expansionoften does not always converge. In fact, anytime a closed-shell molecule is dissociated intoradical fragments the perturbation theory will fail. We call these PT defeating correlations”strong”. The most striking consequence of quantum many-body theory is that particlescan become non-locally ”entangled” with one another. These are precisely the situationswhere no single determinant dominates the exact wave-function and a perturbation analysiswill fail. Often nearly degenerate single-particle spectra are indicative of such a situation.Because this sort of correlation isn’t the same as the local-exclusion, ”dynamical” effect itis also sometimes called a ”static” correlation problem.

A determinantal expansion of the exact wave-function in a strongly correlated situationoften has a number of significant determinants which compete for dominance. The numberof such determinants grows exponentially with the number of strongly correlated particles,and forms the fascinating crux of the strong correlation problem. These are often related toeach other by greater-than-double replacements, and accompanied by the very large numberof small-weight configurations which model the dynamical exclusion of electrons from eachother. There are however models capable of capturing these strong correlation effects, towhich we turn our attention. We note in passing that our problem has a converse: stronglycorrelated quantum systems can describe exponential complexity with linear numbers ofparticles. This is the force behind the growing field of quantum computation.

1.1.4 The Complete Active Space Self-Consistent Field

FCI isn’t affordable for very many electrons, but usually the strong correlations of amolecular system occur between just a few single-particle states and the remaining are well

1.1. CONTEXT 9

described by the Hartree-Fock model. To exploit this fact we can solve CI within a spaceof only a few orbitals, and make those orbitals well-defined by introducing a variationalcondition which determines them. The model which results from solving a CI in an ”activespace” of only a few electrons and orbitals and variationally minimizing the correlationenergy with respect to perturbations of these orbitals is called multiple configuration SCF(MCSCF) [46–48]. If the CI is chosen to be complete the resulting model is called aComplete Active Space Self-Consistent Field (CASSCF) model.

|Ψ〉CASSCF =∑

C∈Active

C|0〉 (1.26)

|0〉 = U |Ψ〉HF where U = eθ (1.27)

s.t.δE

δC= 0, and

δE

δθ= 0 (1.28)

(1.29)

Because the CI wave-function obeys a variational principle, the gradient is given triviallyby the Hellman-Feynman Theorem.

δEMCSCF

δU= 〈Ψ|δH

δU|Ψ〉MCSCF (1.30)

Starting with the HF MO’s a CASSCF wavefunction is determined by solving the active-space CI, forming the orbital gradient, taking a step in θ and transforming the Hamiltonianinto the new basis until convergence. In addition to the occupied-virtual rotations whichmust be optimized in HF, rotations must be performed between active and inactive occu-pieds and virtuals. θ is a non-linear parameter meaning that there may be more than oneset of orbitals satisfying the CASSCF condition which could be chosen to model a givenelectronic state. Still it is common to proceed as if there were a unique solution givena number of orbitals chosen to be active, and a number of electrons chosen to fill thoseorbitals. Admitting an active occupied and virtual orbital for each valence electron pair ofa molecule is often an excellent choice, and called the perfect-pairing valence active space.When it is computable for the complete valence space the CASSCF wavefunction providesa smooth and qualitatively correct model of electronic structure, but too often the smallactive space which is affordable is not accurate.

Aside from the exponential scaling which permanently limits its range of applicabil-ity, CASSCF also systematically underestimates the correlation energy because it largelymisses dynamical correlation. In most cases it has such significant overlap with the ex-act wave-function that a perturbative correction for these missing correlations is largelysuccessful [47]. This divide-and-conquer approach to the correlation problem is a commonelement of most multi-reference correlation methods.

1.1. CONTEXT 10

1.1.5 The Coupled Cluster Anstaze

Truncating the CI excitation operator at any level of excitation results in a model whoseerrors grow with system size, and for chemistry where particle number changes in the courseof a reaction this is extremely undesirable. In statistical mechanics the exponential formof the partition function is motivated by the extensivity of the energy. Likewise if we wantan extensive wave-function we should choose an exponential shape [49–51]. Suppose anansatze of exponentiated cluster operators up to rank n :

ΨCC〉 = eT |0〉 (1.31)

T =∑i=1,n

Tn and Tn =∑i,b

T b1,b2...bni1,i2...ina†bn ...a

†b1ai1 ...ain (1.32)

This is the trial wave-function of a coupled cluster model. These models are amongstthe most successful and accurate in quantum chemistry [52] and strongly influence themodels developed in this thesis, although they were first developed to model the interactionsbetween nucleons [53]. To solve for the parameters of the expression above, one might betempted to introduce the dual of the ket, and solve using the Ritz variational method [54].Unfortunately even if T is much smaller than the span of the Full-CI vector to do sorequires an exponential amount of computational effort. An alternative set of non-linearhomogeneous equations for each parameter can be derived if the ansatze above is insertedinto the SE and projected on the left with each level of excitation (µi) present in the clusteroperator:

HΨCC〉 = EeT |0〉 → 〈µi|e−T HeT |0〉 = 0 (1.33)

With this prescription the computational cost of a coupled-cluster model is the same as thesimilarly truncated CI,O(m2n+2) wherem is the dimension of the basis and n the rank of theexcitation operator), but the resulting model is extensive and significantly more accurate.Unfortunately the price paid is the loss of the variational lower-bound. Most moleculesat their equilibrium geometry can be quantitatively described if the cluster operator istruncated at triples. The coupled-cluster singles-doubles model (CCSD) is a ubiquitousand highly successful variant. This expansion of the wave-function makes no differentiationbetween weak-or-strong correlations, but fails if the truncation rank of the cluster operatoris smaller than the number of strongly correlated electrons in the molecule. At the time thisthesis is written computer resources are such that CCSD(T) [55] is the method-of-choiceif a system can be represented with roughly fewer than 2000 basis functions, and thesemethods afford chemical accuracy if there is no strong correlation problem. Local variantswith near-linear scaling have been developed [56, 57], although we cannot yet reach theregime where that linear scaling is cost-effective.

The cluster operator is isometric and so by similarity transforming the Hamiltonian

1.1. CONTEXT 11

e−T HeT we do not alter its spectrum, although the spectral expansion of each state isshifted downwards into the lower-rank blocks of the transformed Hamiltonian matrix. Thetransformed Hamiltonian can be diagonalized, just like H itself providing correlated excitedstates which are usefully accurate for states dominated by a single excitation, even at thelevel of doubles. This is called Equation of Motion (EOM)-CC [58, 59]. However thistransformation does create an effective Hamiltonian which is non-Hermitian, and that hassome non-trivial computational and physical implications which are beyond our scope.

1.1.6 Orbital-Optimized CC as an approximation of CASSCF

CC is a size-extensive truncation of CI, and so one might wonder whether CC with anorbital optimization condition would also make a useful and size-extensive truncation ofCASSCF. This direction was pursued by the Head-Gordon group roughly a decade ago. Theresulting Valence-Optimized Doubles model is defined by a pseudo-variational Lagrangianand orbital optimization condition:

Ec = 〈0|(1 + Λ)e−T |H|eT0〉 (1.34)

Where: Λn =∑

i1...in,b1...bn

Λb1...bni1...in

a†in ...a†i1ab1 ...abn (1.35)

dE

dT= 0→ (Multiplier Condition) (1.36)

dE

dΛ= 0→ (Amplitude Condition) (1.37)

dE

dθqp= 〈0|(1 + Λ)e−T |[H, (aqp − apq)]|eT0〉 = 0 (1.38)

(1.39)

When the strong valence correlations have the character of two doublets coupled to a sin-glet the VOD wave-function is an accurate approximation to CASSCF for non-interactingsystems. The computational costs of VOD grow with the sixth order of molecular size,which is significantly smaller than NMO!, but if more than 1 electron pair is strongly corre-lated (the dissociation of multiple bonds. metals, etc.), VOD is an unsatisfactory model ofthe electronic structure. The 6th order effort required for these non-local models precludestheir application to large systems and they still lack dynamical correlation.

1.1.7 The Perfect Pairing Model

Bonding electron pairs are a useful idea [60] which have found ubiquitous applicationsin chemistry since the invention of a Lewis dot-structure. This indicates that bonding pairsmay also be a useful way to concisely express a many electron wave-function. The minimal

1.1. CONTEXT 12

model which is exact for non-interacting electron pairs is called the Generalized-Valence-Bond Perfect Pairing [61,62] method. The ansatze takes the form:

Ψpp〉 =∑i

T i∗ i∗

ii |0〉 (1.40)

where i, i, i∗, i∗ are an occupied alpha, occupied beta, virtual alpha and virtual beta or-bital respectively. These ”pair” quartets of spin-orbitals are uniquely associated with oneanother by rotating the orbitals self-consistently amongst themselves. This requires addi-tional rotations beyond CASSCF within each space, because there is no longer invariancebetween occupied rotations. One can solve the anstaze introduced above as a truncatedcoupled-cluster doubles model [63,64] using the same equations presented above for VOD,except only allowing each amplitude and multiplier to possess indices of one electron pair.This approximate wave-function can be afforded for hundreds of electrons and usuallycaptures between 30 and 60 percent of the valence correlation energy. Unfortunately PPexaggerates the locality of electron correlations, resulting in symmetry-breaking artifactsfor systems with multiple resonance structures (benzene, allyl radical etc.). It has been thepurpose of this thesis to find generalizations of this idea which provide improved accuracy,approaching the accuracy of CASSCF, and finally approximate the total electronic energy.

Models Beyond PP

To address the symmetry breaking problem our group has pursued several directionsbefore this thesis. Attempts to reduce symmetry breaking by allowing 2-pair excitations(in the doubles space) resulted in the Imperfect-Pairing (IP) [65–67] model, which capturessignificantly more correlation energy but still exhibits most symmetry artifacts. The IPmodel’s correlation energy also unphysically diverges for dissociation problems. This wasrepaired by introducing a truncation of the amplitude equations inspired by GVB (GVB-RCC) [68] which removes the offending terms, but is still physically lacking for multi-bond dissociation processes. Later perturbative three-pair amplitudes were combined in ahybrid Lagrangian from the IP model. The resulting three-center imperfect-pairing model(TIP) [69, 70] improved correlation energy recovery into the 90% range for most systems.A somewhat different approach attempted to add the missing projective information fromgreater-than doubles by truncating the extended-coupled cluster model. The resultingquadratic coupled-cluster doubles [71, 72] models improved on the performance of CCSDfor dissociation problems, at the price of increased cost.

Dynamical Corrections

Parallel to work developing an affordable reference for strong correlations our groupdeveloped corrections which afford accurate total energies by adding back in the largenumbers of small correlation contributions which arise from orbitals outside of the valence

1.1. CONTEXT 13

virtuals. These corrections [73–77] were developed by introducing a matrix-based Lowdin[45] style partitioning of the coupled cluster Hamiltonian, in a spirit much like MP2. Thenumerical performance of these models relies entirely on the strength of the reference. Incases where that reference is solid, but expensive (like CCSD), the resulting models areremarkably accurate, but expensive (7th order).

1.1.8 The Elephant in the room: DFT

This thesis also describes some work on density functional models of chemistry. Theseapproaches are significantly cheaper than any other correlation method and can afford nearchemical accuracy, given a simple electronic structure and empirical fitting procedures.However they achieve this efficiency by giving up the systematic improvability which is thefeature of the ab-inito models described above. In fact they largely ignore the non-localstrong correlation physics which are the thrust of this thesis.

The founding theorems [78] of this formalism state that the ground state density some-how uniquely determines the ground state energy although the functional dependence ofthe energy on the density is not constructively established. For a one-electron system it’srelatively elementary to derive the exact functional (since there is a simple map betweendensity and wavefunction), but for many electron systems the exact functional is not avail-able, unless the system is otherwise exactly soluble. To continue we postulate that for eachinteracting many-electron system there exists a continuous map (an adiabatic connection)to a fictitious non-interacting system under a local external potential which possesses thesame density as the true physical density. In the case of Kohn-Sham [78–81] DFT, thenoninteracting system we imagine to be the HF wave-function, resulting in an effectiveHF-like equation for the fictitious orbitals:(

h+ Tdiff + J + Vxcρ(r))φi(r) = εiφi(r) (1.41)

Vxcρ(r) ≈∫δrρ(r)F (ρ(r),∇ρ(r), ...) (1.42)

Here Tdiff is the difference between the kinetic energies of interacting and non-interactingsystems (assumed 0 in our case) and Vxc is a functional of the electron density absorbingall exchange and correlation effects. The second line above reflects the functional formwhich is applied in the vast majority of all DFT calculations. Note that this implies anapproximation that the exchange energy is semi-local (which it is not). The largest errorsof most functionals come about from the missing cancellation between J , and it’s anti-symmetric partner, K which causes electrons to repel themselves.

To construct models for Vxc, early work focused on analytically soluble model systems,like the uniform electron gas (UEG). As the functional is made more-and-more generalthese approaches have been superseded in chemistry by empirical fits of functional forms

1.2. OUTLINE OF THIS WORK 14

against large sets of standardized chemical data [82,83], sometimes preserving the uniformelectron gas limit. This empiricism affords accuracy, especially for thermochemistry whichis dominated by local, dynamical correlations.

So there is a possible connection between the electron density and the true energy ofthe ground state. Of course we can say something about how this functional must behavefor toy systems. It depends on the distributions of electrons everywhere, and cannot bewritten as a semi-local gradient expansion of ρ(r). Even supposing that it was some integralover two electron positions: Exc ≈

∫δr1δr2F (ρ(r1), ρ(r2)) (and this is not so!) the cost

of such a construction would grow with the size of a molecule like MP2. No functionalin common use is exact for even a single-electron atom. If we had an expression for theexact functional, such an expression would be intractable for large systems, and likely nomore appropriate a starting point than FCI. Nonetheless DFT is an enormously successfulmodel of dynamical correlations in chemistry. No thesis towards the correlation problem iscomplete without some understanding of how DFT achieves this massive feat.

1.2 Outline of this Work

1.2.1 A Sparse Automation of Many-Fermion Algebra

Algorithms useful in the construction of electron correlation models are collected along-side new developments for cases of high rank and sparsity. In the first part of this paper aBrandow diagram manipulation program is presented. The complementary second sectiondescribes a general-rank sparse contraction algorithm which exploits the permutationalsymmetries of many-fermion quantities. Several recently published local correlation mod-els (perfect quadruples and perfect hextuples) were built using these codes. This papershould facilitate reproduction and extension of high-rank electron correlation models thatcombine truncation by level of substitution with truncation by locality, such as the num-ber of entangled electron pairs. This Chapter has been published as a paper in MolecularPhysics [84].

1.2.2 The numerical condition of electron correlation theorieswhen only active pairs of electrons are spin-unrestricted

The use of spin-unrestriction with high-quality correlation theory, like coupled-cluster(CC) methods, is a common practice necessary to obtain high quality potential energysurfaces. While this typically is a useful approach, we find that in the unrestricted limit ofROHF fragments the CC equations are singular if only the strongly correlated electrons areconsidered. Unstable amplitudes which don’t represent the physics of the problem are easilyfound and could be unwittingly accepted without inspection. We use CCD stability analysisand the condition number of the coupled-cluster doubles Jacobian matrix to examine the


problem, and present results for several molecular systems with a variety of unrestrictedcluster models. Finally a regularization of the CC equations is proposed which allows usto apply CC, and Lagrangian gradient formulas even with completely unrestricted orbitals.This Chapter has been published as a paper in The Journal of Chemical Physics [85].

1.2.3 The Perfect Quadruples Model

A local approximation to the Schrodinger equation in a valence active-space is suggested,based on coupled cluster (CC) theory. Working in a pairing active space with one virtualorbital per occupied orbital, this Perfect Quadruples (PQ) model is defined such that elec-trons are strongly correlated up to ”four-at-a-time” in up to 2 different (occupied-virtual)electron pairs. This is a truncation of CC theory with up to quadruple substitutions (CCS-DTQ) in the active space, such that the retained amplitudes in PQ are proportional to thefourth root of the number of CCSDTQ amplitudes. Despite the apparently drastic natureof the PQ truncation, in the cases examined this model is a very accurate approximation toComplete Active Space Self-Consistent Field (CASSCF). Examples include deformationsof square H4, dissociation of two single bonds (water), a double bond (ethene), and a triplebond (nitrogen). The computational scaling of the model (4th order with molecule size)is less than integral transformation, so relatively large systems can be addressed with im-proved accuracy relative to earlier methods such as perfect and imperfect pairing which aretruncations of CCSD in an active space. This Chapter has been published as a paper inThe Journal of Chemical Physics [86].

1.2.4 The Perfect Hextuples Model

We present the next stage in a hierarchy of local approximations to complete activespace self-consistent field model (CASSCF) in an active space of one active orbital per ac-tive electron, based on the valence orbital-optimized coupled-cluster (VOO-CC) formalism.Following the perfect pairing (PP) model, which is exact for a single electron pair and ex-tensive, and the perfect quadruples (PQ) model, which is exact for two pairs, we introducethe perfect hextuples (PH) model, which is exact for three pairs. PH is an approximationto the VOO-CC method truncated at hextuples containing all correlations between threeelectron pairs. While VOO-CCDTQ56 requires computational effort scaling with the 14th

power of molecular size, PH requires only 6th power effort. Our implementation also in-troduces some techniques which reduce the scaling to fifth order, and has been applied toactive spaces roughly twice the size of the CASSCF limit. Because PH explicitly correlatesup to six electrons at a time, it can faithfully model the static correlations of moleculeswith up to triple bonds in a size-consistent fashion and for organic reactions usually repro-duces CASSCF with chemical accuracy. The convergence of the PP, PQ, PH hierarchy isdemonstrated on a variety of examples including symmetry breaking in benzene, the Cope


rearrangement, the Bergman reaction and the dissociation of fluorine. This Chapter hasbeen submitted as a paper in The Journal of Chemical Physics.

1.2.5 Dynamical Correlations: The +SD correction

The multi-reference cluster approach based on single-reference formalism(SRMRCC) iscombined with paired, active space treatments of static correlation to produce a satisfyinglysimple cluster truncation amenable to strongly correlated problems. An implementationof the method is compared to benchmark results for F2 and H2O dissociation problems,the H4 and H8 model systems, and the insertion of beryllium into hydrogen. The modeldemonstrates the simplicity, accuracy and compactness offered by orbital-optimized coupledcluster models (OO-CC), and the possibility of a local method for strong correlation. ThisChapter has been submitted as a paper in The Journal of Chemical Physics.

1.2.6 A Density Functional Aside

The exchange energy of a uniform electron gas which experiences a novel 2-parameterseparation of the Coulomb interaction is derived as a local functional of the electron density.The 2 parameter range separating function allows separate control of where and how rapidlythe Coulomb interaction is switched off, as opposed to conventional 1-parameter errorfunction attenuators. The usefulness of the functional is briefly assessed by combinationwith a recently published pair of exchange and correlation functionals. The self-interactionerror (SIE) of noble-gas dimer dissociation is found to be reduced while thermochemistryis relatively unperturbed. These results suggest that attenuator shape is a direction bywhich range-separated exchange functionals may be further improved. This Chapter hasbeen published as a paper in Chemical Physics Letters [87].

17

Chapter 2

Many Fermion Algebra

2.1 Introduction

There are no mysteries in the first-principles of quantum chemistry, but many prob-lems in realizing these principles as usable models. Often the premise of an idea can beexpressed easily, but the path to code overwhelms human endurance. Shifting the effortof many-Fermion algebra onto computers is not a new idea [88]. Some fundamental earlytechniques like string-based configuration interaction [89, 90] already shifted symbolic ef-forts onto the computer. These ideas had some early impacts exploring the performanceof high-orders of many-body perturbation theory [91]. Extensions of [92–94] similar ideaswere later used to create exponential operator strings and consequently an (unfactorized)Coupled-Cluster(CC) code. Alongside these developments the diagrammatic bookkeepingtechniques [95] which are so useful many-body theory were pushed to their limits [96, 97].

A symbol manipulation program requires an investment of time, and it isn’t much of an”end in itself”, and so until recent years these tools have been uncommon. The general CCprogram of Kallay and Surjan [98], was a substantial step forwards. Near the same timean ambitious project to use symbolic algebra to generate optimized correlation models be-gan [99]. In recent years interesting demonstrations of these techniques have begun to yieldmodels which would not have been imaginable without them. In particular it became pos-sible to formulate multi-reference state-specific cluster theories of very high-order [98,100],some local static correlation models [101], and implement explicitly correlated cluster mod-els [102] which approach spectroscopic accuracy amongst other applications [59, 103, 104].Outside of the quantum chemical community similar ideas are also being pursued [105].

In this paper we document a framework for the automated derivation and subsequentnumerical evaluation of many-Fermion expressions. The algorithms presented feature a di-agrammatic (Antisymmetrized-Brandow [106]) formalism which yields explicit equations,implicit treatment of permutational symmetry and a general contraction algorithm. Thecontraction algorithm introduced leverages sparsity on its arguments in a general way. This

2.2. MANY-ELECTRON ALGEBRA 18

novel feature is especially important for high-rank correlation models, which would oth-erwise be simply too expensive for all but the smallest applications. Virtually any sortof localization scheme can be exploited, and we demonstrate that the cost of the result-ing algorithm is insensitive to tensor rank. Subjects which have been the focus of otherwork [107–109] (factorization and loop structure) are not investigated, but the ideas de-scribed in those treatments can be leveraged in this scheme directly [108]. This work alsopresents the details which make our recently published local correlation models [101] re-producible, and hopefully extensible by other groups.

2.2 Many-Electron Algebra

Wick’s theorem [27], which governs the construction of virtually any electron correlationmodel can be coded as a few recursive replacement rules: the Fermion anti-commutationrelations. In Mathematica this can be achieved in less than 10 lines of code, and in prin-ciple it is enough to determine algebraic expressions for matrix elements of any operator.Beginning with sums over every index of a string of operators, Wick’s theorem exploits a re-ordering of the creation and annihilation operators so that the terms which vanish becausea†pa

†p = 0 or apap = 0 can be easily eliminated. δ functions which restrict (”contract”) two

indices on different operators to be the same are introduced by the repeated applicationof the anti-commutators. In quantum chemistry ”contraction” sometimes means makingthese restricted sum expressions and often means evaluating these restricted sum expres-sions. We will describe algorithms for both problems.

With a simple recursive sort implementation of Wick’s theorem as described above onewould not get very far because an exponential number of algebraically equivalent termsare generated by such a procedure. We offer the ten-line Wick’s theorem in the supportinginformation to illustrate that. If one desires a truly general formalism which can handlehigh-rank tensors, contractions must be represented in a topologically unique way, onlyrecording the pattern of connectivity between the tensors introduced by the δ functions.This amounts to employing a diagrammatic formalism, where the δ functions become linesbetween vertices. This work is based on the antisymmetrized Brandow-type diagramswhich dominate the literature of coupled cluster theory and have been described manytimes [52,110]. The resulting models are built on spin-orbitals.

Many readers may be unfamiliar with diagrams altogether, and this paper cannot re-place the introductory reviews mentioned above, but we can try to express the basicidea [26, 111] which is conceptually simple. The four sorts of operators that occur in asecond quantized picture of electrons are mapped onto 4 sorts of directed lines drawn awayfrom a vertex. Creation (a†) lines are drawn above (up) and annihilation (a) lines are drawndown. If the operator involves a virtual (a†b or ab) the line is given an arrow going upwards.

Correspondingly if an occupied level is involved (a†j or aj, the arrow is downwards. At the


NameVaccum

LinesAttached

Operator 1 ....

Total Lines

Attached Operator Notation: NameLines

AttachingTotal Lines

Line Notation: # Up Holes# Down Holes

# Up Particles

# Down Particles

∑ck

T ck Vacik → R1, Denoted: V , (1, 0, 1, 0), (1, 1, 1, 1), (T1, (1, 0, 1, 0), (1, 0, 1, 0))

Where: T1 = T ck = T cka†cak

(2.1)

Figure 2.1: An example of the diagram notation. Each box is a list of 4 integers (line notation)characterizing the normal-ordered many-fermion operator.

base of the diagram you imagine the reference |ket〉 and above the 〈bra|. The advantageof the diagram is that the ”contractions” between indices, which can be written so manyways in a formula, are just one connection between matching arrows. The whole elegantnotation is somewhat isomorphic to the popular children’s toy called ”Lego” where thearrows are the part of the brick which holds on to the one below.

Kallay and Surjan [112] introduced a very clear and compact notation for a diagram ofthe coupled cluster amplitude equations, which strongly influences the notation employedin this work. Operators are described by a name, a sequence of 4 integers (the number ofup and down hole and particle lines they possess), and another quartet for the number oflines connecting to the Fermi vacuum (determinant |0〉). Diagrams are described by listsof operators. Inside each operator the other tensors to which this tensor is connected arelisted by name, and another quartet of numbers for each sort of connection between them.Figure 1 depicts this process. A diagram can be written more than one way with thisscheme, but contraction is fast.

Given a string of normal ordered Fermion operators represented in this fashion a matrixelement between two vacuums can be obtained from the complete set of unique diagramsformed by the constituent operators as described in Algorithm 1. Unique in this contextmeans: ”Has the same number of connections of each sort between each pair tensors”. Thisset is built up in the obvious fashion: operator-by-operator and line-by-line. Algorithm 1is best thought of pictorially. Imagine two piles of Lego bricks left and right, which corre-spond to summed operators H = (Foo + Voooo....) and eT = T1 + T2 + T1T2 + .... We are

going to make the expressions for HeTc which are all the unique structures that sit an H


brick on top of a brick from the right. Select a brick a from the left pile (say Voovv). Nextwe pick up a brick from the right pile, say T1. Voovv has an open down virtual which canconnect with T1’s open up virtual when we place V on top of T . We connect those bricksand set them aside. We grab another copy of Voovv from the left pile and another copy ofT1 from the right, then ask if there are any other connections we could sit H on T1. Thereare not any such new connections and so we pick up a new brick from the right and repeatthe process until there are no new bricks on the right or left. In the supporting informationAlgorithm 1 is given as a Mathematica routine called Kontract[].

for Diagram i ∈ Pool2 dofor Line type t ∈ (o†, o, v†, v) do

if a has open t line and i has t† open thenAttach a with i by t and append to output

end ifend forTake unique union of pool and growing output;

end forAlgorithm 1: Contraction of Diagram a with Pool2

2.2.1 Coupled-Cluster Theory

The coupled-cluster(CC) theory has distinguished itself [52] amongst approaches tothe many-electron problem. This paper will apply the algorithm above to automate thederivation and implementation of these equations. It is useful to cast CC theory in a pseudo-variational language [58, 113, 114] even though it is not economical to optimize eT |0〉 as avariational trial wave-function [115] without the projective ansatze. In order to derive theprojective amplitude equations from a variational principle we introduce a de-excitationoperator Λ = λa...i... a

†i ...aa, as a Lagrange multiplier for each amplitude equation. Given a

Lagrangian pseudo-energy Ec = 〈0|(1 + Λ)HeTc|0〉 a set of non-linear equations for Tcan be derived by assuming this Lagrangian is stationary to variations in Λ (equation 2).The notation c means that T must share at least one line with H if the diagram is to be

included, and HeTc = e−T HeT .

∂Ec

∂Λ= 〈0|(ai...a†a)HeTc|0〉 = 0 (2.2)

We have already walked through the generation of diagrams for HeTc. To make an am-plitude equation from this pool we simply select any diagrams which would be closed whenprojected against the desired excitation manifold. In the notation of the code describedin Figure 1, this means selecting any diagram whose vacuum lines (the second quartet of


indices) are that manifold, for singles: 1, 0, 1, 0, for doubles: 2, 0, 2, 0 etc.. A routinethat performs these manipulations to produce the CC amplitude diagrams with the Kon-tract[] routine is also given in the supporting information as AmplDiag[].

With HeTc in-hand generation of the coupled cluster Λ(multiplier) diagrams, effectivedensity matrices [113], Jacobian [59] etc. are straightforward matters of making the correctpools of diagrams and operating this algorithm on them. For example, let us consider howto generate the Λ equations which arise from the following:

0 =dEcdT→ 0 = 〈0|(1 + Λ)[HeTc, a†p...aq]0〉 (2.3)

H is contracted with eT to construct HeTc, as described above and symbolically in

algorithm 2. Imagining the ”pile of bricks”, we have made for HeTc we now need tosit (1 + Λ) on top of all the pieces in this pile in every unique way. The algorithm (2)is made a little simpler to write down by noting at most quadratic amplitudes and linearmultipliers occur in Eq(3). In the language of the brick analogy it means that our eT pilecontains structures with at most 4 bricks. We don’t need to worry about the 5 brick pieceT1T1T1T1T1 because it cannot make a connected diagram with H that only has 4 lines totouch. So we make every unique product of amplitudes up to the 4th order in the first partof algorithm 2, contract these with H and keep only the unique ones. Then we loop overpossible Λn(0 ≤ n ≤ N) and contract each. This pool is finally then projected on the rightfor the desired excitation level of multiplier, ie: 0, 1, 0, 1 for the Λ1 equation etc. Thisalgorithm is given in the supporting information as LamDiag[]

One need not generate the CC Lagrangian, (1 + Λ)HeTc, in its entirety to constructthe multiplier diagrams, and for very high-rank equations that might not be advisable.We have coded and described it in this way because with this quantity in hand mostinteresting CC equations [116, 117] are just projections against the appropriate manifold.Generalization to analytic second derivatives is also possible [118]. For some responseproperties it is useful to write the Lagrangian [58, 119] as a dot product between H andeffective one(γqp) and two (Γrspq) particle density matrices. In our brick analogy, we look at

every structure from (1 + Λ)HeTc which has no open lines; meaning it contributes to theenergy when we put the reference on either side. We can see the hamiltonian brick inside.If we pull that hamiltonian brick out, the shell we leave behind is Γ, which now has thelines which were shared with the Hamiltonian open.

Ec = F qp γ

qp + V rs

pq Γrspq (2.4)

To obtain diagrams for those density matrices from (1 + Λ)HeTc, one only needs to firstselect the diagrams which are closed on both ends, order them by the H block they contain(if desired), and then delete the H operator and leave the lines dangling to the vacuum.This is also automated in the supporting information in routine: DMDiag[].

2.3. TENSOR REPRESENTATION AND PERMUTATIONAL SYMMETRY 22

Converting a pool of diagrams into the usual strings of sums over basis functions is

Create all disconnected diagrams: T p1n1T p2n2

T p3n3T p4n4

s.t. 0 ≤ pi ≤ N and∑pi ≤ 4

for diagram hi ∈ H dofor diagram Dj ∈ amplitude pool do

Contract(hi, Dj)Discard disconnected diagramsTake unique union of results and growing HeTc.

end forend forfor Λk, 0 ≤ k ≤ N do

for diagram Dj ∈ HeTc do

Contract(Λk, Dj) and gather to form (1 + Λ)HeTcend for

end forfor 0 ≤ l ≤ N do

Project (1 + Λ)HeTc onto |µl〉 (the Λl residual equation)end for

Algorithm 2: Construction of Multiplier Diagrams, rank N

ironically much more tedious to automate, but can be important for the beginner. Tobegin, construct a skeletal list of tensors with dummy indices. Then for each internal lineon each internal tensor collapse two dummy indices from the appropriate tensors into asingle summed index. Sign is easily accomplished by listing the second-quantized versionof a diagram and counting the number of intersections between lines connecting the sameindex. In Mathematica, one can just call Signature[] on a list of indices. The factorarising from the number of equivalent tensors/lines is computed after the diagram has beenexpressed as indexed tensors because its easier to establish subgroups of tensors and indices.Permutation symbols aren’t generated at all because anti-symmetry of tensors is assumedby the contraction algorithm. In the supporting information these tasks are performed byDtA[] with some examples. It remains to describe how the resulting sums of high ranktensors will be achieved, and we now turn to this question.

2.3 Tensor Representation and Permutational Sym-

metry

Computational performance is heavily dependent on the way data is arranged in mem-ory [107]. Ideally contraction can be mapped onto matrix multiplication with minimalreordering, and for situations where tensors may represented as simple arrays in memorythere are codes which can optimize tiling for peak performance [108,120]. However we are

2.3. TENSOR REPRESENTATION AND PERMUTATIONAL SYMMETRY 23

interested in situations where dense representations of data are not tractable, and we alsoneed to deal with high permutational symmetry. One solution is to represent tensors asstrings of second quantized operators [90]. In this way symmetry concerns are easily dealtwith, but vectorization is sacrificed. We make a similar compromise, but choose a slightlydifferent format: a data structure which handles its own symmetries.

For the lesser ranks there are some standard formats [121] interfaced to sparse linearalgebra packages which generalize the common matrix formats. However for high ranksthere is an ordering problem which results if there is some ”leading dimension” which canbe iterated over more rapidly than the others. Our first approach was to introduce ann-dimensional net in which each node knew the location of its nonzero neighbors in eachdimension. Sparse contraction can then be effected quite simply by traveling in the correctdirection along this net. Memory bandwidth limited the capability of this idea insofar aswe implemented it because element addition requires traversing each dimension of the gridto find the appropriate niche. Nevertheless this structure was used to initially debug theamplitude equations of the PQ [86] model.

If a sparsity pattern exists within the data because of spatial locality [56,122,123] or apair constraint [101, 124–126], or some other local model [127] this must also be leveragedin the storage scheme. Desiring the most general code possible, we adopt the simple sparsecoordinate representation: i1, i2, ...,Value. The Fermion index symmetries are easily ex-ploited in this representation by representing only those coordinate-value pairs which aresorted with respect to an ordering on these symmetries. Each tensor data object is givenan attached symmetry object which lists the sets of dimensions which are antisymmetric toone another. Because they are now a natural part of the data structure, contraction actu-ally becomes simpler because there is no need to determine lengthy permutation symbols.Before the evaluation of a contraction the symmetries of the two argument tensors are usedto establish the symmetry of the result if the result is an intermediate. If the result is anamplitude, these symmetries are already known, and only the unique output coordinatesobeying the exclusion principle are incremented.

The storage of the coordinates adds substantial memory overhead, but in the mostgeneral case there is no alternative. The costs of these symbolic operations are poised todecrease, as computer scientists develop optimized, vectorized [128, 129] and parallelizedsort algorithms. Random-access writes are made cheapest with a hash-map data structure.The data is usually written to a simple array when it must be sorted for the algorithms ofthe next section. As development in high-rank methods continues, tensor decompositionapproximations will likely be used by ourselves and others to reduce the cost of storage andcontraction.

2.4. CONTRACTION 24

2.4 Contraction

We assume a pair-wise factorization of each term of the electron correlation theory, bothbecause this has been given excellent attention by other authors [108,120] and because onemay achieve the ”correct” cost exponent with virtually any pairwise factorization of anonlinear contraction. An example is useful, and so consider this term of the CCSDTQ Λ2

residual equation:

1

16Λa,b,p11,p12i,j,h7,h8

T p9,p10,p11,p12h5,h6,h7,h8V p9,p10h5,h6

→ 1

16Λa,b,p11,p12i,j,h7,h8

Ip11,p12h7,h8→ Λab

ij . (2.5)

We will use the contraction of Λ (called Tensor 1) with the intermediate (Tensor 2) asour example. Now to complete the calculation of an electron correlation model all thatremains is to evaluate an anti-symmetrized sum over internal dimensions with matchingindices. An anti-symmetrized sum means that the Λab

ij must be antisymmetric with respectto permutations of the index labels (because it represents a Fermionic wavefunction), andour contraction algorithm is made responsible for ensuring this. It quickly becomes impos-sible to perform the products which are identical to each other by antisymmetry more thanonce because these grow exponentially with tensor rank. We will now discuss how thesecan be avoided using the storage scheme from the previous section. We often must talkabout whether an index is summed over or mapped onto the result. The former are called”internal” and the latter ”external”.

Ignoring anti-symmetry the ”contraction” itself is uniquely defined (Alg number, linenumber) [Alg.3:1] by a signed factor, C = 1

16, a matched list of contracted dimensions

(dk1, dk2), and another which determines the source dimension of a result dimension (rk, sk).If we number the dimensions of a tensor: T n+1,n+2...

0,1,...n then these lists would be denoted:(2, 0), (3, 1), (6, 2), (7, 3) and (0,Λ0), (1,Λ1), (2,Λ4), (3,Λ5) in our example. Becauseonly elements with sorted indices have been stored and the other elements related to theseby permutation are implied, the dimensions in these lists must be permuted for the symme-tries of each argument. This process is described generally in algorithm 3 and specificallyfor our example below.

In our example both tensors have anti-symmetries between the contracted occupiedand virtual indices. If we simply applied every permutation of both tensors to (dk1, dk2)some multiplications would be performed more than once. For example the permutationsIp11,p12h7,h8

= Ip12,p11h8,h7and Λa,b,p11,p12

i,j,h7,h8= Λa,b,p12,p11

i,j,h8,h7would only permute the contraction list and

give, (3, 1), (2, 0), (7, 3), (6, 2). Likewise there are several (dk1, dk2) which would be emptybecause of the sorting on the indices, like (2, 1), (3, 0), (6, 2), (7, 3). In order to leveragesymmetry we must maintain the order of dk1 and dk2.

For the special case of the CC equations this step is simplified by the fact that onetensor’s symmetries will always contain the symmetries of the other. For the sake of theroutine that follows assume that the symmetries of dk2 map to a subset of the symme-

2.4. CONTRACTION 25

tries on dk1 (as they do in our example) and in the other situation reverse the labeling.First, obtain the number of redundant contractions from tensor 1 and multiply the factorby the inverse [Alg. 3:9]. This number, Order(S2), is the product of the factorials ofthe number of contracted dimensions which fall in each symmetrical set. In our examplethere are two occupieds which are contracted and antisymmetrical to one another and like-wise two virtuals so the factor is 1

2!2!. Next, apply all the permutational symmetries of

tensor 1, denoted σk, to dk1 and keep only those results ck1 which are ordered, adjust-ing the sign accordingly. For example beginning with dk1 = 2, 3, 6, 7 the permutationσ1 · Λa,b,p11,p12

i,j,h7,h8→ (−1) ∗ Λa,p11,b,p12

i,j,h7,h8produces another unique ck1 = 2, 3, 5, 7 [Alg. 3:9].

Often σ produces a redundant ck1, and these are not added to the contraction list [Alg3: 6,16]. When this process is finished for the example there are 36 unique ck1 out of the576 permutations which exist on Λ4. Likewise apply all the symmetries of tensor 2 to dk2and keep only those which are ordered in tensor 2 and map to a ck1 which is sorted in theorder of tensor 1, adjusting the sign accordingly [Alg. 3:19]. In our example there is onlyone such dk2: 0, 1, 2, 3. It’s immediately obvious how important the symmetries are.We would be doing 576*4 = 2304 loops in the non-symmetric algorithm, instead of 36. Ifour algorithm ignored symmetry and stored the tensors in a redundant fashion we wouldbe wasting the same amount of effort. Given loops over the lists of contracted dimensionsgenerated by algorithm 3, ck1 and ck2, symmetry has been taken care of, and exploited.We need only evaluate a simple sum when these contracted dimensions match and sort theindex of the result, which makes the greater task of sparse contraction somewhat simpler.

In local electron correlation models [57,122,130] amplitudes are so sparse that the poly-nomial scaling exponent of cost with respect to system size can often be decreased by severalorders if floating point effort is not squandered on zero multiplications. One must somehowidentify the nonzero sets of matching internal dimensions, dk1 and dk2, on each partnerwithout explicitly looping over both tensors or assuming every possible dk1 is nonzero.In yet another phrasing, alignment must be achieved between the internal dimensions ofthe two contracted tensors to realize sparse contraction. Previous work on rank 3 and 4tensors has employed sorting [131] or simulated annealing [57] to achieve this alignment.Let’s make this concrete by returning to the example:∑

h7,h8,p11,p12

1

16Λa,b,p11,p12i,j,h7,h8

Ip11,p12h7,h8(2.6)

In a non-sparse correlation model this sum would be achieved as four nested loops over thesummed indices, and an 8th order amount of effort would be expended regardless of thenumber of nonzero elements. A first guess at the sparse analogue might be to loop overnonzero Λ and nonzero I, performing multiplication when they match on h7, h8, p11, p12.This algorithm’s cost grows with the product of number of nonzero entries in each tensor,even if there are no non-zero multiplications to be performed. If the number of nontrivialelements in each grows in a greater-than-linear fashion, then this algorithm will also grow

2.4. CONTRACTION 26

Input: (C, dk1, dk2, (rk, sk))for σ1 ∈ One’s anti-symmetries do

3: Permute ck1 = σ1 · dk1if ck1 disobeys sorting of One or Two then

Continue loop6: else if ck1 ∈ result then

Continue loopelse

9: add (Sign(σ1) COrder(S2)

, ck1), to result list 1.end if

end for12: for σ2 ∈ Two’s anti-symmetries do

Permute ck2 = σ2 · dk2if ck2 disobeys sorting of Two then

15: Continue loopelse if ck2 ∈ result then

Continue loop18: else

add (Sign(σ2), ck2), to result list 2.end if

21: end forAlgorithm 3: Exploitation of Permutational Symmetry

unphysically with system size. However if the two tensors are given an order before thesummation, then by following the elements in sequence we can establish if there is a non-zero partner without any effort.

The indices of nonzero elements on the dimensions h7, h8, p11, p12 in Λ and I, calledck1 and ck2 respectively, which we initially have stored as a simple unordered list, aresorted in a lexical order. Suppose that dk1 was 2, 3, 6, 7 as for our example. Then afterthis sorting the list of elements representing Λ are strictly ordered as: (X,X, 1, 2, X,X, 3, 4) <(X,X, 1, 2, X,X, 3, 5) < (X,X, 6, 2, X,X, 3, 5) etc. This ordering is exploited by the con-traction code to avoid a nested loop over the elements of both tensors, as described in theinner while() loop of algorithm 4. The loop begins with two pointers at the least elementsof both tensors. If the two elements do not match on the contracted dimensions the loopadvances the pointer which lexically precedes the other on the contracted dimensions andotherwise the floating point effort of contraction is performed as usual. By constructionthe number of comparison/skip operations cannot exceed the sum of the number of nonzeroelements in both tensors. Regardless of the sparsity pattern this means that the cost ofalgorithm 4 only grows with the number of non-trivial products in each argument. In thesupporting information the inner while() loop is implemented for simple coordinate tensors

2.4. CONTRACTION 27

as a C++ routine, SparseContract(). Quantum chemists are used to high-performancematrix algebra packages, but there is a growing literature of computer science devoted tomore symbolic tasks [132] like sorting. Some codes have even been developed to performthese operations on highly vectorized graphics processors, and these should be exploited infuture work.

At this point most of the notation in the algorithm 4 has been described. We also use ashorthand for the ck indices of the ith element in a block : cki . For example if Block1

were two elements: (7, 6, 1, 2, 8, 9, 3, 5), (3, 1, 6, 2, 9, 0, 3, 5) and ck1 was 2, 3, 6, 7 thenck11 is 1, 2, 3, 5. It is often useful to separate the elements of the tensor into blocksbecause large numbers of indices are incompatible with one another, for example becauseof spatial or spin symmetries. Algorithm 4 mentions this idea because in the applicationwhich follows we have enforced a sparsity on tensors by blocking them into local groups oforbitals. We assume that the reader can group elements of tensors into blocks based onthe dimensions which are contracted, and can decide whether two blocks will contribute tothe result with some compatibility rule.

for each ck1 from Algorithm 3 doBlock One on dk1for each ck2 from Algorithm 3 do

Block Two on ck2for Block1 ∈ Blocks of One do

Extract all compatible elements of Two into Block2.(Note: Block1, Block2 raster sorted on ck1, ck2)Index in Block1, i1 = 0 and Index in Block2 i2 = 0while i1 < Size(Block1) and i2 < Size(Block2) do

if ck1i1 < ck2i2 (Raster sense of <) then++i1

else if ck1i1 > ck2i2 then++i2

else if ck1i1 = ck2i2 thenSort result index and keep the sign.Result += (Value at i1)*(Value at i2)*(Sign from index sort)*C*(Sign andsymmetry factor from ck1, ck2)

end ifend while

end forend for

end forAlgorithm 4: Sparse contraction loop

2.5. CONCLUSIONS 28

2.4.1 Rank insensitivity

To give a brief demonstration of the usefulness of this contraction algorithm considerthe following term which occurs in the CC/CI equations of any order ≥ 2:

T a3,...an,b1,b2i1,...in

〈a1b1||b2, a2〉 → Ra1,...an

i1,...in(2.7)

The naive cost of such a contraction is O(onvn+2), and so already at quadruples this termmakes CC intractable for all but the smallest systems. The feature of the contractionalgorithm developed in this section is that the CPU cost of this term can be limited bydefining some subsets of E ⊂ i1, ...in, a1, ...an, I ⊂ b1, b2 and then the CPU cost of thecontraction is limited to O(E ∗I). In our pair-based models [86,125], we limit the growth ofthese subsets using a particular labeling, but the contraction algorithm only requires thatone be able to quickly compute if an index belongs in E or I, so we could imagine labelingthem by spatial domain etc. For the purposes of demonstration we will limit the growthof E, I with system size to some simple polynomial with a pair-labeling. The indicesof the amplitudes, integral and residual are each spanned by a certain number of mutuallyexclusive orbital sets oα, oβ, vα, vβ which together span an entire orbital active space.

Supposing that T , R and V are restrained to two pairs, the scaling of this contractionis reduced to third order for n ≥ 2. In the three-pair case this summation takes on at most4 pair labels, and so it ideally scales with the 4th order of system size regardless of rank.For perspective we provide the size of the three-pair amplitudes in Figure 2. The nonzerodimensions of all amplitude ranks between 2 and 6 vary over roughly an order of magnitudebetween the (22,22) and (40,40) active spaces (where (X,Y) denotes the space formed by Xelectrons in Y orbitals). For a growing number of pairs we separately timed this contractionfor amplitudes between doubles and hextuples with a three-pair approximation.

As the rank grows, the pre-factor due to coordinate storage grows, as does the cost ofsorting T and V on b1, b2 for each permutationally unique contraction. Furthermore thesorting cost is not strictly linear; it is sub-quadratic (NLogN). Still using algorithm 4 weachieve virtually the same cost-exponent for the hextuples as we do for the doubles (Figure3). Interestingly, the performance is less predictable in the large-space, high-rank regionof the plot. We can say with certainty this is not due to any peculiarities of a chemicalsystem (all tensors have been filled with noise).

2.5 Conclusions

The computational resources available to scientists interested in physical modeling con-tinue to grow in a non-linear fashion (albeit now in parallel), and so there are ever-moreinteresting applications of computable theories. However the number of man-hours whichcan be devoted to such projects are much the same today as they were a long time ago,and automation of the relevant mathematics will continue to spur new developments. To

2.5. CONCLUSIONS 29

y = 3.214x + 0.059

y = 3.230x - 2.115

2

2.5

3

3.5

4

4.5

5

5.5

1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65

Log

(Num

er o

f A

mp

litud

es)

Log( Active Occupieds )

T2

T3

T4

T5

T6

Linear(T3)

Linear(T6)

Figure 2.2: Number of (symmetrically unique) elements in Tn for 2 ≤ n ≤ 6 for the calculationsperformed in Figure 3 and linear least-square fits. The number of amplitudes are constrainedto grow in a cubic fashion for any rank with a three-pair constraint. Note that Dim(T2) =Dim(〈a1b1||b2, a2〉) since a PP-active space is chosen

y = 4.62x - 6.778

y = 4.786x - 6.606

-0.7

-0.2

0.3

0.8

1.3

1.8

1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65

Log

( Wal

l Sec

ond

s )

Log( Active Occupieds )

T2 T3 T4 T5 T6 Linear(T2) Linear(T6)

Figure 2.3: Wall time consumed in calculating equation 4 for various 2 ≤ n ≤ 6 using algorithm 4,and the cubically growing tensors of Figure 2. Horizontal axis is the number of occupied orbitals,and the space has the same number of virtuals. The cost exponent for n = 2 and n = 6 areestimated by the slope of the linear fits.

2.5. CONCLUSIONS 30

further reduce the amount of time spent by others seeking to reproduce these results weinclude the routines described in Mathematica/C++ code as we have implemented them.Extensions of these routines to other sorts of diagrams that integrate-out spin, or extendednormal ordering [133] to generate a multi-reference cluster theory would be quite usefuland interesting.

The algorithms presented were conceived with an emphasis on generality, and the goalof correct scaling of cost (up to an exponent). They have proven useful for prototypingelectron correlation theories which would otherwise be intractable, but a large additionalpre-factor could be recovered through close attention to the introduction and re-use of inter-mediates, memory layout and vectorization. The growing divide between highly symbolicand multilinear correlation models, and the standard libraries at a computer scientist’sdisposal will need to be bridged.

It would also be quite valuable to develop a general scheme for representing sparse,high-rank data without the memory bandwidth costs of the coordinate representation. Ifthe sparsity has at least some structure (like a pair-constraint), then these can be used toaddress loops, but such a structure is not always available or so simple to order. Multi-linearcompression technologies [134–136] should prove useful in this respect.

31

Chapter 3

Numerical Stability of UnrestrictedCorrelation Models

3.1 Introduction

Many electron models often sidestep the complicated structure of separating electronpairs by allowing approximate wavefunctions to break spin symmetry. The spin-unrestrictedwavefunction [28] has the strength that it matches the energy of non-interacting molecularfragments in the limit of dissociation. As a consequence, spin-unrestricted Hartree-Fockhas become a standard reference for many sorts of correlated treatments including high-quality coupled-cluster (CC) methods [52]. When correlated models are used to optimizeunrestricted orbitals [137] strong correlations between paired electrons are described re-dundantly, and the two competing descriptions can cause difficulties. Multiple solutionsare one manifestation, and another which we have unfortunately encountered while devel-oping orbital-optimized cluster models [138] is singular behavior of the amplitude equations.

Because of their great physical impact, the existence and character of solutions to thecoupled cluster equations are of interest in themselves. Despite the non-linearity and highdimension of these equations much is now known about their solutions thanks to the effortsof several groups [139–146]. Our focus in this paper is much more quotidian, we simplyexplore why we were unable to combine valence-space CC with completely unrestrictedorbitals. We find, to our surprise, that the CC equations are quite generally singular in thevalence space if the one-particle basis is totally spin-polarized in an ROHF-sense, meaningthat the occupied space is comprised of ”valence” spin-orbitals perfectly localized on frag-ments and ”core” spin-orbitals whose α and β parts are identical.

Poor numerical condition of the CC amplitude equations at restricted dissociation isnot new to any practitioner, but with unrestricted orbitals they are usually well-behaved.

3.2. RESULTS 32

This case is noteworthy for a few reasons: it is a general feature of combining unrestrictedorbitals for dissociation with a correlation model in the unrestricted space, it is easy to find”false solutions”, and we can offer a possible solution. The Jacobian and closely relatedstability matrix [147] of the CC equations will be examined for these purposes. The formerhas been examined before for the case of restricted linear [140] and multi-reference [142]cluster theories.

Our analysis begins with a curious set of calculations on the N2 molecule. Orbitals wereprepared such that the only the 6 valence electrons were unrestricted and localized on eachfragment so that a dissociation curve could be followed inwards from the correct asymp-tote at a separation of a few Angstrom. We attempted to apply CCD in the active space,and with simple amplitude iterations convergence was sluggish. Inspection of the ampli-tudes revealed that they were a scaled unit vector. Employing a standard DIIS [148] solverthe correction vector was zeroed in a few iterations, but again the amplitudes appearedunphysical. The gradient obtained from these amplitudes took strange orbital optimiza-tion steps, and the same results were found independently in our two totally independentimplementations of the theory, and so we proceeded to examine them further.

3.2 Results

In all that follows CC calculations were performed in the active space formed by thepairs relevant to a bond-dissociation process. The orbitals were unrestricted such thateach fragment reproduces the ROHF limit, this will be called the unrestricted limit. It isimportant to stress that our conclusions only hold for an cluster model where all occupiedspin-orbitals are localized on high-spin fragments. They do not hold for the usual UHF or-bitals because of the spin-polarization of pairs which would be restricted in the ROHF case.

Many attempts were made to obtain a solution by continuation. At N-N separationsof less than 1.7A the S2 of the reference determinant is well below the spin-polarized limit(3) even for simple Hartree-Fock orbitals and the coupled cluster equations can be solvedeasily, but this solution cannot be followed to the dissociation limit. Even at the dissocia-tion limit, one can easily solve the CC equations if a single term linear in the amplitudesis neglected. We attempted to continue this solution by introducing a simple continuousdeformation parameter λ as the coefficient of a term and solving the equations along thereal axis between λ = [0, 1], but were unsuccessful. Convergence stagnated, but no singleelement of the amplitude diverged. A similar situation has been observed in some otherstudies of singular CC equations [141]. The same attempts were made for several otherdissociation problems (Ethene, H2, etc.) with the same results. Further analysis of homo-topies [143–146] can completely characterize the solutions of non-linear equations shouldthey exist, and can establish the precise identity of a non-linear singularity (pole, branch,

3.2. RESULTS 33

pinch etc.), but any such distinction is of mathematical, not physical, concern as was es-tablished the pioneering work of others [139] (c.f. Section IV(d)). We will instead focuson firmly establishing the singularity of the Jacobian, the scope of the problem, develop-ing a similar set of well-conditioned equations, and heuristic understanding of this situation.

The Jacobian characterizes the response of the coupled cluster equations to a linearperturbation:

∂2E

∂T∂Λ=∂〈Φβγ

uv |HeTc|Φ0〉∂tλµlm

. (3.1)

This matrix is the size of the amplitude vector squared; if one of it’s eigenvalues shouldbecome non-positive either the equations have no well-behaved solution, or solutions meetat this point. In either of the previous cases the amplitude vector to which the the Ja-cobian belongs should not be regarded a good approximation to a physical ground state.Furthermore one may say that as a linear approximation of an amplitude iteration step, the”condition number” (the ratio of the Jacobian’s smallest and largest eigenvalues) measureshow rapidly the equations can be solved by iteration. This non-hermetian matrix has beenderived and coded into programs many times [149–151] in the history of quantum chem-istry. For the purposes of this paper we produced a computer-generated implementation,as have others [152, 153]. The results of the automatic implementation are complementedby a separate program derived and coded by hand for the cases of our local cluster modelswith which all stability matrix calculations were performed. In all cases the Jacobian wasexplicitly constructed and diagonalized to avoid the art of guess construction.

Figure X depicts the lowest eigenvalue of the CC Jacobian for the case of ethene disso-ciation as obtained by explicit diagonalization. The orbitals were prepared at the ROHFdissociation limit, then allowed to restrict as the fragments coalesce. Several sets of ampli-tudes were examined: the null guess (alternatively this can be considered the Linear Cou-pled Cluster (LCC) Jacobian), the MP2 guess, the best amplitudes which can be reachedby simple iteration (as seen in the figure), those produced by DIIS, and even amplitudesiterated from random noise, all to the same effect. The increase of the lowest eigenvalue onthe inclusion of single excitations is noteworthy. Performing the same exercise for analo-gous dissociation problems produces the same results. By 5A the condition numbers of allof these dissociation processes are so large that convergence seems impossible with doubleprecision arithmetic. Of the cases examined Mo2 is the most stable, with a smallest Jaco-bian eigenvalue of roughly 1 ∗ 10−4 at 7A.

If the active space is expanded so that not all the orbitals are spin-polarized, the equa-tions become immediately well-behaved; the resulting Jacobian eigenvalues are strictlypositive at any displacement. The reader is undoubtedly familiar with the reasonablecondition of UCC calculations and so this should be evidence enough. Having converged

3.2. RESULTS 34

-0.1

0

0.1

0.2

0.3

0.4

0.5

2 3 4 5 6 7 8

Internuclear Distance (Angstrom)

Sm

alle

st E

igen

valu

e (e

V)

Ethene, CCD (4,4)Ethene, CCSD (4,4)MoMo, CCD (12,12)N-N, CCD (6,6)

Figure 3.1: Smallest Jacobian eigenvalue for various fragment dissociations, in the 6-31G basisfor all cases except Mo2.

these amplitudes for the case of ethene (12,12), they were projected on the completelyunrestricted (4,4) active space as a guess (the orbitals are unchanged between these twocalculations only the amplitude space) and the singularity remains whether one tries toconverge from this guess, or immediately diagonalizes the Jacobian. Based on these resultsand the previous observations we argue that the singularity can be understood with thelinear part of the Jacobian which does not depend on the amplitudes. Inspection of thefragment orbitals provides another simple argument, all non-linear CCD terms depend onintegrals of the sort 〈oo‖vv〉 and for these fragment localized spin-orbitals (Figure X) theseare separated across space and vanishing.

The coupled cluster stability matrix contains information very similar to the Jacobian,but can be used to understand the convergence properties of the iterative process we relyupon to solve these equations. Surjan [147] and coworkers recently published work exam-ining this matrix for several solutions of the CC equations along dissociation curves. Theirresults showed that the CC equations may exhibit a diverse range of iterative behavior(convergence, chaos and divergence) if manipulated by a denominator shift, and that ex-trapolative methods like DIIS [148] can be misleading as they seem to converge on whatappear to be stable fixed points. We reiterate the formulas for this matrix given CCD andthe usual partitioning. One can see that it is essentially the Jacobian dressed by factorswhich reflect the conventional form we use to solve the CC equations.

3.3. MODIFIED EQUATIONS WHICH ARE WELL-CONDITIONED 35

J22uvβγ,lmλµ = δulδvmδβλδγµ −

∂〈Φβγuv |HeTc|Φ0〉/∂tλµlmfββ + fγγ − fuu − f vv

(3.2)

The Lyapunov exponents are the central object of this analysis, the logarithm of the Sta-bility Matrix’s eigenvalues. If these should equal or exceed zero iterations are not con-vergent. We turned to this tool because we wanted to understand what was occurringwhen simple iterations would stagnate at very small residual values. Table X depictsthe results for several small molecules obtained with a hybrid of our local CC methods:The Perfect Pairing (TPP =

∑ti∗i∗

iia†i∗ a

†i∗ aiai) [61, 137, 154, 155], and Imperfect Pairing

(TIP = TPP +∑

i 6=j ti∗j∗

ij a†j∗ a†i∗ aj ai + tj

∗i∗

ij a†i∗ a†j∗ aj ai) [65, 66, 68, 124] models. Note that even

if the species is asymmetrical the same trend is observed.

These results firmly establish the singularity of the CC equations in the unrestrictedlimit, but what meaning, if any, does this have for UCC as it is practiced when completespin polarization is almost never the case? We can imagine one situation when this shouldbe kept in mind, if the restricted core electrons are cut from a calculation by a pseudo-potential the results are essentially the same as those which would be obtained from theminimal active space. Figure X depicts the CCD Jacobian’s lowest eigenvalue for the dis-sociation of Mo2 with the CRENBS basis and matching pseudo-potential.

3.3 Modified Equations Which are Well-Conditioned

Since the instability and ill-conditioning of the CC amplitude equations at the unre-stricted limit has been established, we seek to restore solubility to these cluster modelswhen the appropriate ROHF fragment orbitals are desired. Here we will discuss a fewpossible solutions based on the idea of regularization [156]. We will rate them based on asimple set of criteria: a) does it stabilize the CC amplitudes, b) is it simple to define andimplement, c) how heavily does it affect the energetics of the molecular system at equilib-rium, and d) will it allow us to optimize orbitals with active space CC Lagrangian methods.

The first and simplest correction is to add a constant denominator shift. As shownby Surjan et al. [147], this enables us to make the amplitude equations stable. However,it requires a constant shift of at least 12 kcal/mol to be able to optimize the orbitalsalong the entire ethene dissociation curve. This is a very strong penalty near equilibriumbond lengths where the amplitude equations are usually well-conditioned. We also tooka non-linear equation solving approach and tried to identify the source of the singular-ity, eliminate it, solve the system when it is non-singular, and follow a homotopy backclosely [157]. In the section below we attribute the singularity to the structure of the block

3.3. MODIFIED EQUATIONS WHICH ARE WELL-CONDITIONED 36

1 2 3 4 5 6Bond Length

-0.6

-0.4

-0.2

0

Expo

nent

CH2CH2CH2CHFN2

(a)

3.5 4 4.5 5 5.5 6Bond Length

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0

Expo

nent

CH2CH2CH2CHFN2

(b)

Figure 1: The change in the largest Lyapunov exponent for a stability analysis PP and IPexchange type amplitudes for 3 example molecules using the IP energy Ansatz on the PPorbitals in a minimal active space with the 6-31G* basis (a). (b) is an enhancement aroundthe unrestriction point.

1

Figure 3.2: Lyapunov Exponents along the Dissociation Coordinate.

of the linear coupling matrix containing the PP and the IP exchange type amplitudes,and created a stable nonlinear system and solution by eliminating the off-diagonal matrixelements. The CC amplitude equations are easily solved in that diagonal representation,and the homotopy can be followed in very closely to the original problem. However the ho-motopy could only be followed in to a scaling parameter of at maximum around 80.0% forthe IP level of correlation, and it is prohibitively expensive to follow the homotopy properly.

Another approach can be taken from our recent work [70]. A denominator shift remi-niscent to amplitude regularization that has the form −γ(e(t/tc)2n − 1) can be constructedto affect the amplitudes similarly to a static shift. Unlike the static denominator shift thepenalty function approach is flexible enough to be very small before the unrestriction point

3.4. HEURISTIC UNDERSTANDING OF THE PROBLEM 37

on the dissociation PES where amplitudes are typically small, and large when amplitudesbecome large (typically on non-variational surfaces the amplitudes are greater than one).Of course to apply a gradient in the presence of such a penalty we must propagate thismodification through the derivatives of the Lagrangian, but the resulting gradient ”works”in the sense that we can use it to optimize orbitals and geometries.

The simple choice for the critical value of t parameter, tc, is one, since that prevents acomplete inversion of the reference and the doubly excited state. The other parameters (γand n) should be chosen to balance making the corrections small at equilibrium, with ensur-ing that orbitals can be optimized towards dissociation. The shift we propose is −8(et

6−1)Eh, and a corresponding shift of −8(et

6(1+6t6)−1) Eh for the Lagrange multipliers. There

is only a cost in correlation energy of 9.1 µ Eh for ethene at its equilibrium geometry withthe IP+DIP method, and a mean absolute error of 11.5 µ Eh for the PP active space G2aand G2b sets [158,159] done with 6-31G*. A choice of a γ of 2 and a power of 4 also worksquite well, but it bears an energy cost around twice as large (both RMS and MAE) toour suggested parameters. Although the CC equations need to be penalized to be solved,this approach seems to be preferable as only modification to the amplitude and lagrangemultipliers need to be made, and there is only a very slight cost in terms of correlationenergy.

3.4 Heuristic Understanding of the Problem

There is a very simple argument for linear dependence in the basis composed of excita-tions from the orbitals of the unrestricted limit. For the usual UHF method at equilibriumone can imagine spin-polarization as a degree of freedom provided by the basis of excita-tions (and orbital rotations), and it is plain that in the unrestricted limit with only thespin-polarized orbitals excitations do not span this degree of freedom but the number ofparameters is the same. One can also ask how this is manifest in the representation ofthe hamiltonian in this basis explicitly, and this is much less straightforwards. The resultsdemonstrate that generally this singularity is present in the coupled cluster equations andstrongly suggest that it lies in the linear part of the Jacobian. There is a possibility thatboth the linear and non-linear equations are singular but for different reasons, but we willassume that their conditions stem from the same problem.

The simplest possible case is the dissociation of a H2 molecule in the minimal STO-3Gbasis. There is only one unique amplitude in the wavefunction, and one unique orbitalunrestriction parameter which reflects the rotation of the beta bonding MO into the betaanti-bonding MO. The coupled cluster equations for the amplitude have the simple formof a quadratic equation, whose coefficients can be constructed for any distance, R and any

3.4. HEURISTIC UNDERSTANDING OF THE PROBLEM 38

1.0 1.5 2.0 2.5 3.00

20

40

60

80

Distance, !!"!!Degr

ees"

Energy !Eh "

1.0 1.5 2.0 2.5 3.00

20

40

60

80

Distance, !!"

!!Degr

ees"

Amplitude

"1.0 "0.5 0.0 0.5 1.0Contour Key " Parameters

"1.0 "0.8 "0.6 "0.4 "0.2 0.0Contour Key " Energy

1.0 1.5 2.0 2.5 3.00

20

40

60

80

Distance, !!"

!!Degr

ees"

Constant Term

1.0 1.5 2.0 2.5 3.00

20

40

60

80

Distance, !!"!!Degr

ees"

Linear Term

1.0 1.5 2.0 2.5 3.00

20

40

60

80

Distance, !!"

!!Degr

ees"

Quadratic Term

Figure 3.3: CCD parameters for H2 in STO-3G basis, horizontal axis θ, vertical axis R (A).

orbital rotation, θ, these surfaces are plotted in Figure X.

0 = A+BT + CT 2 (3.3)

First notice on the plot of energy that variational determination of the orbitals wouldnot unrestrict at any distance. The simple 1-amplitude CCD expansion can handle theopen-shell singlet. Next focus on the line passing through 45o, both the constant andquadratic terms vanish, but the linear term doesn’t, so one of the two roots diverges in thisunrestricted limit, while the other goes to zero. We can dissect the linear term further andfind that there is a single diagram responsible for the non-vanishing term with the usualalgebraic form: ∑

kc

< ka||ic > T kjcb ⇒ BT2 (3.4)

In the numbering of Figure X the 2-electron integral that appears in this term is 〈11∗||11

∗〉.The coulomb operator lies between orbitals lying on the same atom, and so it fails to decayas the bond is broken.

Considering a more general case, such as ethene dissociation, the CCD equations arenow of much higher dimension, but we maintain that the essential features of this singularityare the same. The linear part of the equations is now multidimensional, and the coefficientof the linear term is a matrix which is precisely the well-known LCCD Jacobian [140],alternatively the CCD Jacobian for the null guess, also known as the matrix U in thenotation of some other papers [138]. For complete CCD the dimension is too large topermit direct inspection, but we can examine truncated block of amplitudes (Table X)spanned by a local model for physical insight, remembering that they exhibit the samebehavior seen in the full active-space CC models examined in section 2, and were the root

3.5. CONCLUSIONS 39

Figure 3.4: Orbital labeling for H2 dissociation.

t1∗1

∗

11t2

∗1∗

12t1

∗2∗

21t2

∗2∗

22

t1∗1

∗

110.093 -0.047 -0.047 0.000

t2∗1

∗

12-0.047 0.093 0.000 -0.047

t1∗2

∗

21-0.047 0.000 0.093 -0.047

t2∗2

∗

220.000 -0.047 -0.047 0.093

Table 3.1: Linear coupling matrix for the PP and IP exchange amplitudes for ethene (H2C=CH2)at a C-C bond length of 7.50 Awith unrestricted PP orbitals in the minimal active space in the6-31G* basis.

of our interest in this problem. A figure labeling the relevant 1 particle functions is alsoincluded for clarity (Figure X).

Ironically, orbitals that lack any spin-symmetry impart a fragment symmetry onto thismatrix which causes it’s determinant to vanish. The off-diagonal matrix elements are foundto arise in the same diagram examined above and the troublesome 〈ov‖ov〉 integral. Wecan provide further argument for the structure of this matrix for our local models becausethe number of amplitudes is manageable, and general formulas for the determinant giventhis structure, and the interested reader is referred to the appendix.

3.5 Conclusions

Recently an interesting study was directed at the question, ”Do broken-symmetry ref-erences contain more physics than the symmetry adapted ones?” [160]. In that case the

3.6. APPENDIX 40

Figure 3.5: Orbital labeling for ethene dissociation.

concern was for RHF orbitals with broken spatial symmetries, and it was found that it wasdifficult for single-reference CC theories to correct the symmetry-broken wavefunction. Ina similar vein, we have found that in the case of completely broken spin symmetry, thephysics of spin correlation is removed from the cluster equations such that they are singular.This manifests itself in poor numerical condition of the cluster equations which preventsus from finding physical solutions and hampers orbital optimization. Ad-hoc manipulationof the Lagrangian using a penalty of the form −γ(e(t/tc)2n − 1) can render the equationssoluble, even in this situation, and it seems that the remaining correlations are relativelyunperturbed. Philosophically, our results suggest that CC based on UHF references suc-ceeds (in the sense that most dissociation curves are reasonable even in the intermediateregion) largely through the independence of strong spin correlations from others. Thisbodes well for correlation models designed on the principle of dividing these problems.

3.6 Appendix

3.6.1 Singularity in constrained cluster wavefunctions

In the arguments that follow, a pair correspondence between alpha and beta spin orbitalswill be assumed by our notation. We enforce such a correspondence on our constrainedCC models, however we note that by the pairing theorem [161] it is valid to apply suchlanguage to unrestricted wave-functions generally, as they may be cast into this form.

3.6. APPENDIX 41

t1∗1

∗

11t2

∗1∗

12t1

∗2∗

21t2

∗2∗

22

t1∗1

∗

110.102 -0.047 -0.056 0.000

t2∗1

∗

12-0.047 0.102 0.000 -0.056

t1∗2

∗

21-0.056 0.000 0.102 -0.047

t2∗2

∗

220.000 -0.056 -0.047 0.102

Table 3.2: Linear coupling matrix for the PP and IP exchange amplitudes for fluoroethene(HFC=CH2) at a C-C bond length of 7.50 Awith unrestricted PP orbitals in the minimal ac-tive space.

The electronic Hamiltonian is separable into operators affecting each fragment separatelyH = H1 + H2, and because of the high-spin structure of our reference any opposite-spindoubles amplitude doesn’t alter the number of electrons on a fragment (only their spin),and spatial symmetry dictates that the spatial parts of the orbital in each pair should bethe same on both fragments.

The Two Electron-Pair Case.

The situation of the 2-pair case of two-atom dissociation (an active space with 4 spa-tial, 8 spin orbitals) is similar physically to that of the 1-pair case. However, instead oftwo degenerate states that we can use the amplitudes to decide between, there is a largerdegenerate subspace leading to far more zeroes in our Hamiltonian. The couplings betweenthese degenerate states again leads to potentially infinite amplitudes, and the necessity ofsolving the CC equations iteratively leads to quite a problem here. A simple description ofthese degenerate doubly excited references can obtained with the pairing notion, as thereare amplitudes that show coupling within a pair (PP), ie: T 1∗1

∗

11, and opposite spin (OS)

exchange-type amplitudes that couple the two pairs (IP), ie: T 21∗

12. Because the determi-

nants generated by all of these amplitudes only differ from the reference by flipping thespins of two of the spin-orbitals, the diagonal of the doubles-doubles block of the Hamil-tonian is some value, a. Breaking down the off-diagonal matrix element coupling a PPamplitude to an IP amplitude, 〈Φ1∗1

∗

11|H|Φ2∗1

∗

12〉, over the two fragments, we see that on one

fragment the projections of these determinants are the same as generated on the diagonalof H. Because the two determinants differ by 2 spin-orbitals the matrix element is a singleintegral, 〈11

∗|22∗〉 which may be identified with the integral examined in the 1-pair case.

These matrix elements which share a single index may be denoted b. Matrix elements whichshare no indices, ie. PP of one pair with PP of the other, differ by more than two spinorbitals and are thus zero. The result is that the Hamiltonian has the following structure(in the basis: T 1∗1

∗

11, T 2∗1

∗

12, T 1∗2

∗

21, T 2∗2

∗

22):

3.6. APPENDIX 42

t1∗1

∗

11t2

∗1∗

12t3

∗1∗

13t1

∗2∗

21t2

∗2∗

22t3

∗2∗

23t1

∗3∗

31t2

∗3∗

32t3

∗3∗

33

t1∗1

∗

110.142 -0.036 -0.036 -0.036 0.000 0.000 -0.036 0.000 0.000

t2∗1

∗

12-0.036 0.142 -0.036 0.000 -0.036 0.000 0.000 -0.036 0.000

t3∗1

∗

13-0.036 -0.036 0.142 0.000 0.000 -0.036 0.000 0.000 -0.036

t1∗2

∗

21-0.036 0.000 0.000 0.142 -0.036 -0.036 -0.036 0.000 0.000

t2∗2

∗

220.000 -0.036 0.000 -0.036 0.142 -0.036 0.000 -0.036 0.000

t3∗2

∗

230.000 0.000 -0.036 -0.036 -0.036 0.142 0.000 0.000 -0.036

t1∗3

∗

31-0.036 0.000 0.000 -0.036 0.000 0.000 0.142 -0.036 -0.036

t2∗3

∗

320.000 -0.036 0.000 0.000 -0.036 0.000 -0.036 0.142 -0.036

t3∗3

∗

330.000 0.000 -0.036 0.000 0.000 -0.036 -0.036 -0.036 0.142

Table 3.3: Linear coupling matrix for the PP and IP exchange amplitudes for nitrogen (N2) at aN-N bond length of 7.50 Awith unrestricted PP orbitals in the minimal active space in the 6-31G*basis.

H = (H1 + H2) =

a b b 0b a 0 bb 0 a b0 b b a

(3.5)

This matrix is singular if b = −a/2 which is empirically found to be the case, and explainedbelow. Of course this is only a sub-block of the linear coupling matrix, U, which wouldoccur in complete CCD, and we affirm with calculations that in unrestricted CCD as it isusually practiced this is not a concern.

The Many-Pair Case.

In general, for a given PP amplitude there are three classes of possible OS exchangeIP amplitudes: those which share no indices, those which excite from the same α orbital(we will call this one the fragment 1 corresponding IP amplitude) and those which excitefrom the same β orbital (fragment 2). The matrix elements of these combinations wereexamined above. If two IP amplitudes excite from a common index they will also have anonzero matrix element b, and with these rules in hand one may construct H. Consider ablocking of HIJ such that a PP amplitude I and its corresponding fragment 1 IP amplitudes(indexed by J) are grouped together in a block which is ordered by their beta indices. Inconventional spin-orbital notation this ordering is written: T 1∗ i∗

i1 , T 2∗ i∗

i2 , T i∗ i∗

ii , ..., T n∗ i∗

in .Symbolically one may construct the diagonal block (A) for the general case and see that ithas the shape below, and determinant (a− b)n−1(a+ (n−1)b). The off diagonal blocks (B)are themselves diagonal, with value b and determinant bn. The complete matrix has the

3.6. APPENDIX 43

following block structure with this ordering, and the determinant of H for n pairs (n > 2) is(a−2b)(n−1)2(a+(n−2)b)2(n−1)(a+2(n−1)b). The matrix will be singular when b = −a/2,or if (a+ (n− 2)b) = 0 or (a+ 2(n− 1)b) = 0. In the full space, correlations of spin-pairedbasis functions often lift this linear dependence by coupling to this block, explaining therobust strength of unrestricted coupled cluster demonstrated in the literature.

A =

a b · · · bb a · · · b...

. . ....

b b · · · a

; B =

b 0 · · · 00 b · · · 0...

. . ....

0 0 · · · b

; H =

A B · · · BB A · · · B...

. . ....

B B · · · A

(3.6)

44

Chapter 4

The Perfect Quadruples Model

4.1 Introduction

The chemical community benefits greatly from systematically improvable single refer-ence ab initio electronic structure methods that can be applied to molecules when Hartree-Fock (HF) is a valid approximation. Recent advances in computational technology havemade methods such as coupled-cluster (CC) singles and doubles (CCSD) routinely appli-cable to dozens of heavy atoms [56, 57, 162, 163]. Some post-HF methods are becomingappreciably cheaper than HF itself [127, 164], and likewise the cost of hybrid DFT. Whenthe mean-field HF reference doesn’t provide an accurate basis for perturbation theory orlow-rank coupled cluster techniques such as CCSD, the situation is quite different. Themost commonly applied approach for coping with strong correlation is complete activespace self-consistent field (CASSCF) [46, 165], but the growth of cost with the size of thecorrelated space is prohibitive and unphysical considering the dimension of the electronicHamiltonian [166]. Moreover, CASSCF cannot provide chemically accurate energetics, andremains an exponential bottleneck behind dynamically-corrected treatments built upon itlike CASPT2 [167] and multi-reference cluster theories [168–172]. There is a need for atractable, black-box, approximation which can be used in lieu of CASSCF for large sys-tems.

Many clever models have been proposed to overcome the CASSCF bottleneck for theproblem of strongly correlated electrons. Broadly they can be divided into two groups :one that tries to overcome the complexity of the wavefunction by resorting to alternativedescriptions of the system often based on the density matrices (density matrix renormaliza-tion group (DMRG) [173,174], variational reduced density matrix (RDM) [175–177], etc.),and another group that tries to build the strong correlations into an accurate zero orderwavefunction (valence bond [60,63,178,179], the related geminal models [180], renormalizedcluster methods [116], spin-flip models [181–183], etc.). Despite a large body of work nosingle model has become attractive and reliable enough to consider the problem solved, and

4.1. INTRODUCTION 45

the direct consequence is that quantitative studies of reactivity are usually only possiblewhen a handful of important (strongly correlated) electrons can be modeled in a CASSCFcalculation.

The importance of local correlation methods to the future of ab-initio many-electrontheory cannot be exaggerated, because in virtually all cases of chemical interest we al-ready have procedures which, if computable, would be exact. The greatest deficienciesin our understanding of the many electron problem are reflected in the way our physicalexpressions become unbearably complicated if only a handful of new bodies are added. Tocombat this problem local methods were introduced early in the history of quantum chem-istry [122,184–186], and continue to reduce the cost of useful calculations [123,130,162,187].For dynamical correlation problems like dispersion, the locality of the Coulomb interactionhas been appreciated for a long time, but this information is rarely exploited in non-dynamical correlation methods. DMRG is a notable exception, but it is naturally bestsuited to exploit locality in one dimension.

For several years our group has developed a family of orbital-optimized coupled-cluster(OO-CC) [188] models for static correlation. Locality [125] is incorporated into these mod-els naturally through the definition of chemically relevant electron pairs, via a pairingactive space in which each electron pair is described by 2 orbitals, one nominally occu-pied or bonding, and the other nominally empty or antibonding. For a given electronpair, providing one double substitution amplitude for promotion of two electrons from thebonding to the antibonding level defines the perfect pairing model [60, 61, 63, 189], whichis exact within the active space for a single electron pair, and is size-consistent. Providingadditional amplitudes that simultaneously promote two electrons from two different pairsdefines the imperfect pairing (IP) model [1, 65], which recovers a significant fraction ofthe interpair correlation energy. It is also possible to include the pair correlations thattransfer electrons between 2 pairs [125] or couple three different pairs [190], which gives anaccurate approximation to the complete treatment of all active space double substitutions.However, both PP and IP cluster amplitudes are truncated at double substitutions, usuallyrendering these methods unsuitable for more than two simultaneously strongly correlatedelectrons. This can lead to non-variational breakdowns with restricted orbitals. They canbe partially addressed by modifying either the energy expression, as in quadratic coupledcluster doubles [71, 72], or the amplitude equations, as in the GVB-RCC [191] variant ofIP. However, neither of these improved approaches can exactly solve the problem of even4 strongly coupled electrons (e.g. a double bond breaking) in the pairing active space ,because they still correlate only 2 electrons at a time.

Here we lift this restriction, by defining the truncated cluster model that exactly re-produces the wavefunction of two electron pairs coupled to each other. In a paired clusterframework this is possible through the addition of a single quadruple amplitude to coupleeach pair of PP amplitudes. Additionally one requires the triple substitutions that coupleeach pair of amplitudes. The idea is made concrete in the diagram below which enumeratesthe correlations contained in the resulting model, which we term Perfect Quadruples (PQ).

4.2. THE PQ MODEL 46

PP

IP

PQ:

Figure 4.1: Correlations in the PQ approximation. Horizontal lines represent different orbitalpair spaces, occupied below and virtual above.

The PQ model exactly treats the problem of 4 electrons in 4 orbitals (2 pairs) and is size-consistent. By restricting ourselves to include only electron correlations between two pairswith a cluster operator limited at quadruples, we obtain a model which is also balancedfrom a perspective of computational cost: PQ is a truncation of active space CCSDTQwhich as a function of molecular size, M , retains only O(M2) of the O(M8) amplitudes ofthe parent theory. This model is realized and its performance is evaluated below.

4.2 The PQ Model

Coupled cluster theory has a storied history in quantum chemistry that has been wellreviewed in recent literature [52, 110], so we will only make our ansatz clear and refer thereader for further information. The formulas below assume that the 1-particle basis hasbeen divided into active and inactive subspaces in a well-defined way; the orbitals we nameare exclusively active. Occupied orbitals are denoted i, j, ... virtuals are denoted a, b, ... thesymbols p, q, ... represent any sort of index. One begins by expressing the ket of the CCwavefunction (1) as the exponential of an excitation operator operating on the reference,which in this case will be the orbitals of a perfect pairing calculation. It is this exponential


that ensures a size-extensive treatment of correlation as is required for chemical systems.If a complementary de-excitation operator, Λ, is introduced then a set of equations thatdetermine the amplitudes and the gradient may be derived by zeroing the partial derivativesof a quasi-energy Lagrangian, E. If the excitation operator T is truncated at some rankthe expansion of the exponential contributing to an amplitude equation ends after a certainnumber of terms and at most quartic powers of an amplitude occur.

|Ψcc > = eT∣∣ 0 > ; T =

∑n

Tn ; ; Λ =∑n

Λn

Tn = Ta1,a2,...,an

i1,i2,...,in= t

a1,a2,...,an

i1,i2,...,ina†a1...a†an

...ain ...ai1 ; Λa1,a2,...,an

i1,i2,...,in= λ

a1,a2,...,an

i1,i2,...,inain ...ai1 ...a

†a1

...a†an

E =< 0|(1 + Λ)(HeT )c|0 > ;∂E

∂Λ= 0 ;

∂E

∂T= 0

(4.1)

To couple four electrons exactly, amplitudes up to those in T4 must be included, definingthe CCSDTQ method. The resulting triples and quadruples equations are summarizedin (2) below. Complete symbolic expressions for the resulting matrix elements have beenavailable in the literature for many years [96] [97], but their long derivation is equalled by thedifficulty of programming these equations into a computer. For this reason we automatedthe diagrammatic derivation of second quantized expressions as have others [103,109,112].

0 =< µ3|H(T2 + T3 + T4 + T1T2 + T2T3 + T1T3 + T1T4 +1

2!T 2

2 +1

2!T 2

2 T1

+1

2!T 2

1 T2 +1

3!T 3

1 T2 +1

2!T 2

1 T3)|c0 >

0 =< µ4|H(T3 + T4 + T2T3 + T1T3 + T1T4 + T2T4 + T1T2T3 +1

2!T 2

2 +1

2!T 2

3

+1

3!T 3

2 +1

3!T3T

31 +

1

2!T 2

1 T4 +1

2!T 2

1 T3 +1

2!2!T 2

1 T22 +

1

2!T1T

22 )|c0 >

(4.2)

Computing the matrix elements of the second term of the quadruples equations with-out approximation takes O(M10) computational operations, and O(M8) data storage as afunction of molecular size, M . This rapidly becomes impractical for all but the smallestsystems so we adopt a local ansatz based on electron pairs. Within a pairing active space,each pair is described by four orbitals, alpha occupied, alpha virtual, beta occupied, betavirtual, which in principle should be optimized to minimize the total energy. If this is doneconsistently then because orbital variations are nearly redundant with single substitutions,the singles amplitudes become very small. However, if one uses pairing orbitals optimizedat a lower level of theory (for instance we shall employ PP orbitals in the tests reported


later of the PQ model), then single excitations within the active space can serve usefullyas a surrogate for orbital optimization within the active space when the higher level wavefunction differs significantly from the one employed to optimize the orbitals. Therefore weretain single excitations.

Turning to pair-based truncation of CCSDTQ, at least two pairs must be used to con-struct a quadruples amplitude, and retaining the S,D,T,Q substitutions that couple onlyup to pairs of electron pairs defines a quadratic sparsity pattern that we call the PerfectQuadruples (PQ) approximation. This will be the fewest terms necessary to be exact for 4electrons in the pairing active space. Symbolically this means that the spin orbital indicesof an amplitude may originate in at most two distinct pairs as summarized as a diagram(Figure 1) and a definition below (3).

T = T1 + T2 + T3 + T4 ; with Ta1,a2,...,an

i1,i2,...,in:= s.t.ak, ik ⊂ Pair1 × Pair2 (4.3)

The amplitude equations are simply those of CCSDTQ [96, 97] when one neglects all am-plitudes which belong to more than two pairs. If one neglects the triple and quadruplesubstitutions, a two pair approximation is very similar energetically to the Imperfect Pair-ing (IP) [1] [65] wavefunction previously developed in our group, but includes amplitudesthat do not conserve the number of electrons in a pair [125]. These singly and doublyionic amplitudes were omitted from the IP model to avoid the generation of some classesof integrals, but these are included here.

The coupled cluster energy depends on at most double substitutions, and so the triplesand quadruples are necessary only insofar as they correct the doubles amplitudes. It wasrealized some years ago that the poor dissociation behavior of coupled cluster active spacemethods is rooted in their failure to obey a Pair Exclusion Principle [68]. Terms nonlinearin the amplitudes introduce excitations that recouple the spins of pairs to themselves andexaggerate the correlation energy. Deleting these terms as was done in the GVB-RCC modelcan repair the dissociation behavior of the model, but is deleterious near equilibrium. Thequadruples fill this role, but one may imagine that they are turned-on naturally by theamplitude equations when important so equilibrium performance is conserved.

Finally, it is known that low-scaling correlation models based on localized orbitals canbe affected by symmetry breaking artifacts. As the imposed sparsity on the amplitudesis relaxed, these artifacts are lifted by the inherent orbital-invariance of coupled clustertheory at the expense of steeper scaling of computational cost. This local truncation erroris exemplified by the preference of benzene for D3h rather than D6h geometry at the PPand IP levels, a problem which can be almost entirely removed by additionally includingdouble substitutions that couple 3 electron pairs at a time [126]. This local truncationerror leads to a slight energetic preference for localized valence structures over delocalizedones, as in benzene. We mention the role of 3-pair couplings for the sake of completeness,but focus in this publication on the role of the new high-rank excitations. Any chemist

4.3. IMPLEMENTATION 49

can recognize situations where two resonance structures are important to the electronicstructure of a molecule, and these are the cases when the more costly three pair modelshould be employed.

4.3 Implementation

The lengthy coupled cluster equations truncated at quadruples were symbolically de-rived in closed form using a diagram-based symbolic manipulator developed by one ofthe authors (JAP) within the Mathematica package. The framework developed includesroutines to automatically construct C++ codes for the amplitude equations that were in-tegrated in a developmental version of the Q-Chem electronic structure package [192]. Allcontractions are performed as a series of pair-wise contractions with appropriately intro-duced intermediates so that the resulting cluster models scale with the familiar O(M2s+2)rule, where s is the highest retained substitution (4 here). The very large (242) pre-factorassociated with ignoring the antisymmetry of T4 would make the PQ method impracticalfor all but the smallest systems, but the contraction algorithm developed exploits the per-mutational symmetries of all amplitudes and intermediates. The correctness of the coupledcluster implementation was verified by comparing with benchmark results for the CCSDTand CCSDTQ models. Local paired models like PP, IP, and the complete valence-optimizeddoubles were also re-implemented in the course of testing.

A method that preserves only a quadratic number of quadruples amplitudes shouldscale formally much more cheaply than the traditional CCD. To achieve that is a challengebecause sparsity imposed on T4, an eighth rank tensor, must be completely exploited. Tomeet this challenge we have developed a tensor library and contraction algorithm capableof performing these high rank contractions with the correct scaling. A structure is kept onthe tensor such that any subset of dimensions may be fixed and the non-zero entries of theothers may be iterated over without sorting or re-alignment. The effort of maintaining andoperating on the sparse structure is spent entirely during element addition. Convergenceof the amplitudes is accelerated with the DIIS algorithm [193].

The intermediates introduced for the nonlinear terms must also be generated with con-sideration for the sparsity imposed on the amplitude results. Because of the large numberof intermediates this was done by simply limiting the number of pairs on the external di-mensions of the intermediate destined for the result amplitude. This is sufficient to realizeformal M4 scaling with molecule size, which is the same as a local doubles model whereionic couplings are retained. Formally the slowest step of our algorithm for large systemsis the production of the complete set of two-electron integrals in the active space. Table2 collects the average number of floating point operations performed in contraction periteration of the coupled cluster equations for some of the models considered here as imple-

4.3. IMPLEMENTATION 50

y = 5.8433x - 1.36

y = 4.7935x - 0.7848

y = 4.1824x + 0.9198

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9Log(Active Orbitals)

Log(

FLO

PS

/iter

atio

n)

CCSDIPPQ

Figure 4.2: Scaling of methods in the 6-31G basis.

mented in this pilot code. The model system is a set of alkanes of increasing size in thegeometry of an adamantane molecule in a 6-31G basis with full-valence PP active spaces(ie: methane, ethane, etc. ). No spatial symmetry information was employed.

As a pilot implementation it is premature to judge by our program what the limits ofa nearly optimal code would be. Even so, the pilot code is economical enough such thatactive spaces much larger than those usually approachable with CASSCF can be explored inmodest amounts of time and memory. An optimal implementation would employ an integralapproximation such as the resolution of the identity, or Cholesky decomposition. Theseapproximations were recently applied to the CASSCF model [194], however in that casethe exponential cost of solving for the amplitudes in the active space means that integralapproximations cannot significantly extend the range of systems that can be explored.Because only a quadratic number of amplitudes are retained, the PQ model is well suitedto take advantage of integral approximations, and this will most likely be done in futuredevelopment.

4.4. RESULTS 51

4.4 Results

For the model systems considered below, the PQ approximation reproduces valenceCASSCF within less than ≤ 10 kcal/mol of absolute error. Energies obtained parallel theCASSCF results over the entire surface within ≤ 5 kcal/mol. The local pair approximationand the limitation placed on the level of excitation both introduce limited errors into ourmodel. Further improvement depends on disentangling and quantifying these errors forsystems which are well understood. CASSCF calculations were performed with the aid ofthe GAMESS [195] package. Unless otherwise noted the cc-pVDZ basis and perfect pairingoptimized orbitals were employed in the PP valence active space. All models except PPincluded single excitations.

4.4.1 H4

The rectangular arrangement of four hydrogen atoms can exhibit strong static correla-tions [196], with complete configurational degeneracy at the square geometry. It is perhapsthe simplest system for which the local doubles models PP, IP and their parent methodCCD possess severe artifacts: an underestimated correlation energy and unphysical cuspat the square geometry. Note that in the figure CCD was performed without an activespace, and so it includes dynamical correlation whereas the other methods do not. H4 isalso perhaps the simplest model one could construct for a polyradicaloid transition state.The PQ model exactly reproduces CASSCF (4,4) for this system by construction as is seenin Figure 3, while the PP and IP methods yield barriers that are too high by about 20kcal/mol, using unrestricted orbitals.

4.4.2 Water

The simultaneous dissociation of water is a common performance benchmark for modelsof non-dynamical correlation [169]. If only double excitations are included from referencethe restricted coupled cluster expansion becomes qualitatively incorrect away from equilib-rium. In Figure 4 the PQ model is compared with CASSCF, and the same model withouttriples and quadruples (2P) (which is slightly better than the IP model). The correspond-ing energies are shown in Table 1. The cc-pVDZ basis was employed with the restrictedorbitals of the PP wavefunction in the (8,8) active space. Near equilibrium, the doublesmodel and PQ both provide a reasonable approximation to CASSCF. Beginning at 2 A thequality of the PQ approximation is made plain, as it continues to track CASSCF, whilethe 2P doubles curve falls to an incorrect asymptote. If the model were allowed to breakspin symmetry, doubles models would capture the dissociation potential curve qualitatively,though in the intermediate regime they remain quantitatively inaccurate because they lackthe configurations which complete the spin eigenfunction.

4.4. RESULTS 52

-2.15

-2.13

-2.11

-2.09

-2.07

-2.05

-2.03

-2.01

-1.99

-1.97

0.75 0.85 0.95 1.05 1.15 1.25

R(H2-H2) Angstrom

En

erg

y (

Eh

)

CASSCF (4,4)UPPUIPR - PQUCCD (Full Space)

Figure 4.3: Potential energy curve for the rectangular dissociation of H2—H2 with the cc-pVDZbasis and restricted 2-pair doubles orbitals. (H-H distance 1 A)

To decompose the difference relative to CASSCF, the local models are compared inTable 2 against their non-local counterparts CCD, and CCSDTQ within the active space.We want to distinguish local truncation error (due to coupling only 2 pairs) from excitationlevel truncation at quadruples. Within the doubles space, 3 and 4-pair correlations absentin the 2-pair models (and PQ) account for 24 percent of the doubles correlation near equi-librium, which is roughly the same percentage of correlation energy that is missing fromPQ vs. CCSDTQ at equilibrium even though the two pair locality constraint neglects amuch larger fraction of triples and quadruples than doubles. At dissociation the correla-tions missing from the two pair models no longer contribute appreciably to the correlationenergy and the local models both approximate their nonlocal counterparts with ≤ 1mEhof error.

Insofar as correlation energy is additive with respect to the spaces of amplitudes included

4.4. RESULTS 53

R(O–H)A CASSCF(8,8) 2p (8,8) PQ (8,8) CCD CCSDTQ1.1 -76.12104205 0.0087 0.0084 0.0056 0.00481.3 -76.04248739 0.0096 0.0091 0.0059 0.00451.5 -75.96406087 0.0085 0.0076 0.0056 0.00371.7 -75.90366501 0.0061 0.0053 0.0044 0.00231.9 -75.86333002 0.0042 0.0058 0.0028 0.00322.1 -75.83961132 -0.0046 0.0046 -0.0054 0.00282.3 -75.82726866 -0.0161 0.0040 -0.0167 0.00272.5 -75.82129839 -0.0259 0.0036 -0.0263 0.00272.7 -75.81841092 -0.0330 0.0033 -0.0334 0.00272.9 -75.81696701 -0.0380 0.0031 -0.0384 0.00273.1 -75.81622269 -0.0414 0.0030 -0.0418 0.00283.3 -75.81583025 -0.0438 0.0029 -0.0443 0.0028

Table 4.1: CASSCF energies for symmetric water dissociation (Eh), and relative errors of PQ,and non-local active-space models with the RPP orbitals.

(empirically this is often true), one might imagine decomposing the errors of PQ by theirorigin. In Figure 5 the quadruples truncation error (CASSCF - CCSDTQ), doubles localityerror (Error(2P) - Error(CCD)), and triple/quadruple locality error (Error(PQ)-quadruplestruncation error - doubles locality error) are plotted. The encouraging conclusion is thathigher excitations in the valence space are often even more effectively captured by the localapproximation than the doubles. This contrasts with the case of dynamical correlationwhere local models for higher excitations have been a more difficult challenge [197,198]. At2.5 A in the nonlocal quadruples the dimension of T2, T3 and T4 are respectively 328, 1184and 1810, and in the PQ model 100, 48 and 6.

4.4.3 Ethene

The dissociation of ethene and simultaneous dissociation of water are very similar staticcorrelation problems, but in the former case the strongly correlated pairs are nearer to oneanother and the valence correlation space is three times larger. As shown in Figure 6, PQreproduces CASSCF (12,12) faithfully using the RPP orbitals with an NPE of 9 mEh andthe resulting wave-function is spin-pure. Whereas IP falls to an incorrect over-correlatedasymptote. If we allow IP to employ spin-contaminated orbitals it will meet the correctasymptote, but it under-correlates in the intermediate region as will be shown in the nextsection.

4.4. RESULTS 54

R(O–H)A 2p (8,8) PQ (8,8) CCD CCSDTQ Reference Energy1.1 -0.1254 -0.1257 -0.1286 -0.1294 -75.986916721.3 -0.1470 -0.1476 -0.1508 -0.1521 -75.885833791.5 -0.1786 -0.1795 -0.1815 -0.1834 -75.776999341.7 -0.2208 -0.2217 -0.2226 -0.2246 -75.676689941.9 -0.2713 -0.2697 -0.2727 -0.2723 -75.587806512.1 -0.3339 -0.3246 -0.3347 -0.3265 -75.510334992.3 -0.3997 -0.3795 -0.4003 -0.3808 -75.443736472.5 -0.4598 -0.4303 -0.4602 -0.4313 -75.387361582.7 -0.5111 -0.4748 -0.5115 -0.4754 -75.340321522.9 -0.5534 -0.5123 -0.5538 -0.5126 -75.301582913.1 -0.5876 -0.5432 -0.5881 -0.5434 -75.270010323.3 -0.6152 -0.5685 -0.6157 -0.5686 -75.24442247

Table 4.2: Correlation Energies for symmetric water dissociation (Eh) with the RPP orbitals.

R(CH2-CH2) CASSCF (12,12) PQ Error 2P Error CCSD Error1.2 -78.145341 0.0110 0.0112 0.00531.4 -78.179446 0.0109 0.0112 0.00591.6 -78.142116 0.0108 0.0114 0.00651.8 -78.087149 0.0103 0.0115 0.00722 -78.035113 0.0096 0.0115 0.00792.2 -77.993032 0.0090 0.0109 0.00792.4 -77.962413 0.0086 0.0081 0.00562.6 -77.942196 0.0084 0.0022 0.00032.8 -77.929896 0.0083 -0.0049 -0.00653 -77.922803 0.0081 -0.0114 -0.0127NPE 0.0029 0.0225 0.0130

Table 4.3: Correlation energies for dissociation of ethene(Eh) with restricted PP orbitals.

4.4. RESULTS 55

-76.2

-76.15

-76.1

-76.05

-76

-75.95

-75.9

-75.85

-75.8

-75.75

0.9 1.4 1.9 2.4 2.9 3.4

R(O - H) (Angstroms)

En

erg

y (

Hart

ree)

CASSCF(8,8)2P (8,8)PQ (8,8)

Figure 4.4: Potential energy curve for the symmetric dissociation of water with cc-pVDZ basiswith RPP orbitals.

4.4.4 Nitrogen Molecule

The deceptively simple nitrogen molecule is a very rigorous test for models of elec-tron correlation [199], and even very recently continues to be a benchmark for costly andsophisticated [200] models of static correlation [201]. Because it lacks hextuple excita-tions PQ ”turns over” towards an over-correlated asymptote when restricted PP orbitalsare employed. This can likely be ameliorated (without relaxing the spin symmetry con-straint) by either including six-particle components of H or observing a ”Pair ExclusionPrinciple” [191]; we hope to pursue these ideas in future papers, but for now, we employunrestricted orbitals to avoid this issue.

With unrestricted orbitals, PQ should reproduce the entire CASSCF curve qualitatively,and the agreement should be quantitative at the dissociation limits. U-CCSDTQ’s energies

4.4. RESULTS 56

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.9 1.4 1.9 2.4 2.9R(O--H) Angstroms

Rel

ativ

e Er

ror (

Har

tree

)

2 Pair Doubles Locality ErrorQuadruples Locality Error Quadruples Truncation Error

Figure 4.5: Error decomposed approximately by source for symmetric water dissociation withRPP orbitals.

are known to approximate full-CI very well for this system with only a few kcal/mol of NPE.In the highly correlated intermediate regime we may gauge the accuracy of PQ by comparingit against CASSCF. The switch to unrestricted orbitals complicates our assessment of thePQ model because the UPP’s orbitals are more unrestricted (ie. unrestricted at shorterinternuclear distances) than PQ’s optimized orbitals would be. Single excitations assistin this capacity by relaxing the PP orbitals, and the resulting energies compare well withthose of the restricted model before non-variational collapse.

Figure 7 and Table 4 compare PQ and the related doubles models with CASSCF in thevalence PP active space. PQ outperforms the related doubles models, and maintains a verymodest NPE error of only 3.83 kcal/mol (6.1 mEh). Predictably the underestimation ofcorrelation energy is concentrated in the bond breaking region where the hextuples play animportant role in configuration space, but unlike UPP this doesn’t result in an unphysicalbarrier to dissociation. To put this result in perspective, an O(M6) scaling method based

4.5. DISCUSSION AND CONCLUSIONS 57

-78.2

-78.15

-78.1

-78.05

-78

-77.95

-77.9

-77.85

-77.81.2 1.7 2.2 2.7 3.2 3.7

R(CH2--CH2) (Angstroms)

Ener

gy (H

artr

ee)

RPQRPPCASSCF (12,12)RCCSD

Figure 4.6: Electronic energy of ethene (R(C–H) = 1.07 A) in (12,12) space with cc-pVDZ basis

on variational (but approximately N-representable) reduced density matrices was recentlyapplied to this same system in the same basis set and a (6,6) active space with an NPE of18.9 mEh [202]. At 1.8 A in the nonlocal quadruples the dimension of T2, T3 and T4 arerespectively 825, 4360 and 8070, and in the PQ model 165, 80 and 10.

4.5 Discussion and Conclusions

We have introduced a model called Perfect Quadruples (PQ) for static electron corre-lation in a pairing active space. It’s worthwhile to compare PQ with other efficient modelsof static correlation. The recent progress made in DMRG is impressive given the shorthistory of the theory [203] [174], but chemical applications have all employed a decimationoperation which is essentially one-dimensional. The even treatment of excitation regardlessof rank is appealing, and the cost per sweep of the method is on the order of PQ. It will


R[N-N] (A) CASSCF (10, 10) PP IP PQ1.1 -109.12969 0.0876 0.0207 0.00771.2 -109.11145 0.0907 0.0216 0.00691.3 -109.06683 0.0935 0.0229 0.00681.4 -109.0148 0.0958 0.0253 0.00821.5 -108.96273 0.0950 0.0265 0.00891.6 -108.91571 0.0856 0.0252 0.00961.7 -108.87631 0.0682 0.0224 0.01021.8 -108.84984 0.0553 0.0217 0.01291.9 -108.82789 0.0412 0.0165 0.01042 -108.81368 0.0308 0.0125 0.0083

Table 4.4: N2 Total energies (a.u.) and the error relative to CASSCF. (* includes singles) withunrestricted PP orbitals.

-109.20

-109.15

-109.10

-109.05

-109.00

-108.95

-108.90

-108.85

-108.80

-108.75

-108.701.00 1.50 2.00 2.50 3.00

R(N-N) (Angstrom)

Ener

gy (H

artr

ee)

CASSCF (10,10)UPPUPQUIP (With Singles)

Figure 4.7: Unrestricted potential energy curve for the dissociation of N2 in the cc-pVDZ basis.


be exciting to see with further numerical testing how this method performs as a referencefor an efficient total energy calculation. Methods like canonical transformation [204] couldprovide a fair assessment of different references as models for the complete electronic energy.

Two modifications of valence bond theory [180,205] have been recently published whichproduce spin eigenfunctions remarkably with costs comparable to PQ. The valence-bond ap-proaches are physically pleasing because they satisfy a property of the exact wave-function,but their correlation energies are much smaller than CASSCF near equilibrium. Thesemethods represent a significant advance in the way we understand correlated spins. Spin-purity may be important for some applications, but likewise may not be most importantfor reproducing some sorts of data and one may imagine complementary niches for bothVB and MO based methods.

There is also a model based on the spin-flip formalism [206], which describes quadruplyradical systems and their excited states again with roughly the cost of the method justpresented. The spin-flip approach and PQ both treat 4 electron problems well, the formerwith the strength that excited states are trivially obtained, and the drawback that it de-pends on an ROHF reference and doesn’t treat the non-interacting ensembles of stronglycorrelated systems that occur in large molecules.

Of all these methods, PQ is distinguished as the one which most strongly resemblesthe CASSCF approach quantitatively and qualitatively. It is a size-extensive truncation ofCASSCF within the active space, which is exact for two electron pairs in that space. Thismeans that the perfect quadruples model is appropriate for molecules with tetraradicalcharacter and can describe such systems over their entire surface. At the same time,because correlations only connect pairs of electron pairs, the overall cost scales only as(M4) with molecule size. This means all valence electrons can be taken as active withoutthe CASSCF problem of trying to choose a very small (feasible) active space that is stillreasonably accurate. The preliminary implementation and benchmark tests suggest thatthe PQ method is a useful step forward over existing lower rank truncations.

There are many interesting possible extensions, a number of which we hope to developin the future. One conceptually obvious, though practically very challenging extension isto generalize the paired concept to still higher excitations (hextuple excitations coupling 3different electron pairs would be the next member of the family defined by PP and PQ, andwould be exact for 3 strongly coupled electron pairs). Considering appropriate dynamicalcorrelation corrections is obviously important for practical applications requiring chemicalor near-chemical accuracy. Algorithmic improvements that increase efficiency and permitdirect orbital optimization are desirable.

60

Chapter 5

The Perfect Hextuples Model

5.1 Introduction

Quantum chemists distinguish two sorts of correlation, one which is captured easilyby perturbation theory called dynamic, and another called ”static”, or ”non-dynamical”which afflicts molecules whose bonds are nearly broken [168, 196], or which are otherwisepoorly described by methods that depend strongly on a mean-field reference. Quantita-tive ab-initio studies of chemical reactivity are most straightforwardly possible when acomputational method capable of tackling static correlation can be applied to the entiresystem of interest. However the standard approach to static correlation, the complete ac-tive space self-consistent field (CASSCF) method [46,207,208], is exponentially costly as afunction of the number of active electrons. As a result, a number of fundamental reactionsof interest are exceedingly difficult to address with CASSCF, such as the Cope rearrange-ment [209–212], due to the need to choose a truncated active space. A wide variety ofon-going work is directed at developing more feasible methods for treating strongly corre-lated molecular problems [116,174,175,177,178,180,199,202,213–219].

It is desirable to have a systematically improvable hierarchy of approximations for staticcorrelation whose cost increases with accuracy and system size in as tractable a way aspossible, to eliminate CASSCF’s drastic limitations on the number of active orbitals. Inother words, we need active space analogs of the large body of work directed at devel-oping approximations to the Schrodinger equation, as exemplified by the coupled cluster(CC) hierarchy of models: CCSD (truncation at double substitutions from the reference),CCSDT (triples), CCSDTQ etc [52,110,220,221]. One possibility is a hierarchy of valenceactive space orbital-optimized cluster theories (VOO-CC). VOO-CC methods were firstproposed some years ago [188], and implemented at the CCD level, and later extendedto the quadratic CCD model [72]. VOO-CCD is exact for isolated pairs of electrons inthe active space, and is extensive. Higher VOO-CC methods (VOO-CCDT, VOO-CCDTQetc) would constitute a systematically improvable hierarchy of approximations to CASSCF,


as we desire. However, direct implementation of higher VOO-CC methods (VOO-CCDT,VOO-CCDTQ etc) seems unpromising because their scaling with the number of active elec-trons mirrors that of conventional CC theory, and it is possible that even higher excitationsmay be required for strongly correlated problems.

An alternative approach that appears much more promising replaces the VOO-CCDstarting point with one of the simplest models of strong correlations, perfect pairing (PP).The PP model [60,61,179] describes the jth electron pair by two configurations, made fromtwo orbitals, one bonding, φj and one antibonding, φj

∗, and a single amplitude, tj:

Gj = |φjφj〉+ tj|φ∗j φ∗j〉 (5.1)

For a system of multiple electron pairs, the intermediate-normalized PP wave function isthen an antisymmetrized product of the geminal corresponding to each pair, Gj, which isalternatively a local active space variant of CCD [63] where only one-pair amplitudes areretained:

|ΨPP 〉 = exp(TPP )|Φ0〉 (5.2)

Here |Φ0〉 is the reference with all bonding orbitals occupied, and,

TPP |Φ0〉 =∑j

tj|Φj∗j∗

jj〉 (5.3)

where |Φj∗j∗

jj〉 is a doubly substituted determinant where the jth bonding level has been

replaced by the antibonding level, reflecting the second configuration of Eq. (5.1). The PPenergy is minimized with respect to variations in the amplitudes and the orbitals. Thenumber of non-redundant orbital degrees of freedom is increased relative to VOO-CCD,because rotations of the active occupied (or virtual) orbitals amongst themselves also af-fect the PP energy. Within the active space, PP is exact for isolated electron pairs, andextensive, just like VOO-CCD. And, as it is a cluster operator, it can be improved. Avariety of augmentations that include the most important inter-pair correlations at thelevel of double excitations have been suggested [65, 190], which approach the accuracy ofVOO-CCD in the pairing active space at much lower computational cost [222].

How should the PP starting point (exact for a single pair of electrons and extensive)best be extended to more strongly correlated systems in the CC framework? One verypromising approach is to seek exactness for larger numbers of electrons at minimum cost.The next model should achieve exactness for two pairs (4 electrons), meaning that thecluster operator must be truncated only at quadruples (i.e. a version of VOO-CCDTQ).To keep the complexity as low as possible subject to this goal, we chose keep only thoseactive space amplitudes that couple either 1 or 2 electron pairs together. We have recentlyproposed and implemented this idea, which we call the perfect quadruples model (PQ) [86].PQ is an enormous simplification over VOO-CCDTQ which can couple up to 8 different


pairs together. As a function of the number of valence electrons, o, VOO-CCDTQ involvesO(o8) amplitudes and O(o10) computation, while PQ retains only O(o2) amplitudes withO(o4) computation. Numerical tests were generally very encouraging – for instance thePQ absolute error relative to CASSCF(10,10) for N2 dissociation was less than 0.33 eV,while the non-parallelality error was less than 0.2 eV. Of course this performance mustdegrade for problems where the number of strongly correlated electrons increases, whichmakes extensions of PQ desirable.

In this work, we therefore implement and explore the next member of this systematicallyimprovable pair-based hierarchy of approximations to CASSCF. The next level should beexact for isolated systems of 3 electron pairs (6 active electrons), which makes it a subsetof the VOO-CCDTQ56 method. To preserve exactness for 3 pairs, we choose to retainonly those VOO-CCDTQ56 amplitudes that involve 3 electron pairs or fewer. We willrefer to this truncation of VOO-CCDTQ56 as the Perfect Hextuples (PH) model. PH willreduce the O(o12) amplitudes and O(o14) computation of VOO-CCDTQ56 to only O(o3)amplitudes with O(o6) computation. Indeed, as will be discussed later, the computationcan be further reduced to O(o5) with a further approximation. PP, PQ and PH togethercomprise the lowest 3 levels of a hierarchy of pair approximations which cost O(o2p) beforefurther approximations, where p is the number of electron pairs for which we demand exactagreement with CASSCF. Each step of the hierarchy improves upon the previous one intwo respects: first, by inclusion of two additional levels of substitution, and second, by theinclusion of correlations that entangle one additional electron pair at the existing lowerlevel of substitutions.

The restriction on the number of electron pairs that are coupled in the correlation ampli-tudes is essentially a local correlation approximation [122] that is inherent in this hierarchy.The strongest correlations entangle one or relatively few pairs, and balanced truncation byexcitation level (2p) and number of coupled pairs (p) enables the cost to increase so muchmore gradually than truncation by excitation level only. Therefore improved results at eachnew level reflect error reduction from one or both of these two aspects of truncation. It isalso possible to separate the two types of truncation, by separately choosing the maximumlevel, n, of substitutions and the maximum number of different pairs, p whose orbitals areallowed to comprise such an index, to define an (n, p) pairing approximation. We shalloccasionally make use of this additional flexibility. For instance, symmetry breaking inbenzene in PP is known to be a result of neglecting 3-pair double substitutions, ratherthan neglect of higher substitutions [190].

CASSCF itself requires a companion treatment of the neglected dynamical correlation,such as by multireference perturbation theory [64,167,223,224], in order to approach quan-titative accuracy. So too will any approximation in the PP, PQ, PH hierarchy, and thereader may be concerned that it will be more difficult to construct such a model for PH.Even with only a density matrix there are now very promising methods which can providedynamical correlation [204,225–229]. There are further options for PH because we possessa good 0th-order wavefunction and a well-defined partitioning of the Hamiltonian. Our

5.2. THE PERFECT HEXTUPLES MODEL 63

group has developed perturbation theories for active-space cluster models based on Lowdinpartitioning [73], which have been applied to the PP starting point [230]. It is also possibleto reformulate the state-universal multi-reference cluster theory (MRCC) [201] for incom-plete active spaces. It is especially appealing to adapt the state-specific single-referenceMRCC [171] formalism for use with the PP, PQ, PH hierarchy since it produces a unifiedmodel which amounts to an alternative truncation of the CC hierarchy which accounts fortotal correlation, and we will soon publish results along this line.

5.2 The perfect hextuples model

5.2.1 Overview of VOO-CC theory

Our goal is to employ coupled cluster (CC) theory to approximate CASSCF within aperfect pairing active space, optimizing both correlation amplitudes and orbitals [188]. Webegin with the usual CC ansatz which parameterizes a many-electron state, |Ψ〉, as theexponential of a correlation operator, T , acting on a reference determinant, |0〉 (written as

|Φ0〉 in the Introduction): |Ψ〉 = eT |0〉. This correlation operator is restricted to an activespace of orbitals, generally one occupied (in the reference) for each valence electron totalingo and the same number of unoccupied single-particle states. We denote active occupiedorbitals as ik, and active virtual orbitals ak.

Following usual CC theory [52,110], the operator T is chosen to include every possibleorbital substitution, T

a1,a2,...,an

i1,i2,...,in,

Ta1,a2,...,an

i1,i2,...,in= t

a1,a2,...,an

i1,i2,...,ina†a1

...a†anai1 ...ain (5.4)

up to some maximum level of substitution, n. These sets of all possible T for substitutions ofa given level n, are called Tn. The equations that determine the amplitudes corresponding toeach retained active space orbital substitution, ts are obtained by projecting the Schrodinger

equation with the corresponding substituted determinant, 〈µs| = 〈0|(a†a1

...a†asai1 ...ais

)†:

〈µs|e−T HeT |0〉 = 〈µs|HeTc|0〉 = 0 (5.5)

The VOO-CC energy that approximates CASSCF follows from projecting with the refer-ence:

E = 〈0|e−T HeT |0〉 = 〈0|HeTc|0〉 (5.6)

In order to have an energy that can be varied with respect to orbital rotations, it isconvenient to define a pseudo-energy, E that augments the VOO-CC energy, Eq.(6), withLagrange multipliers, λs multiplying each CC amplitude equation, Eq.(5). Defining a de-excitation operator Λ analogous to the excitation operator T , the pseudo-energy can be


written as:E = 〈0|(1 + Λ)e−T HeT |0〉 = 〈0|(1 + Λ)HeTc|0〉 (5.7)

The non-linear equations which determine the amplitudes, energy and gradient can be de-rived by making the pseudo-energy stationary with respect to variations of T and Λ, andfinally with respect to orbital variations, θ:

∂E

∂T= 0 ;

∂E

∂Λ= 0 ;

∂E

∂θ= 0 (5.8)

These θ = θqp(aqp − apq) parametrize a unitary transformation U = eθ, mapping the guess

orbitals to the optimal orbitals H → U †HU . In the traditional OO-CC orbital variationsinclude occupied-virtual mixings (θai ), as well as active-inactive mixings in both the occu-pied and virtual spaces. Equation 8 has some appealing formal consequences: active spaceoptimization is possible; the response theory of the model does not exhibit spurious poles,and some properties are rendered Gauge invariant [231]. However, it should be noted thatif single excitations are neglected [232] Equation 8 does not exactly recover recover theCASSCF energy. Including the singles would require some improved algorithms to obtainorbital convergence in a reliable and practical manner, and as of yet this problem remainsunsolved, but can be side-stepped if the orbital gradient is modified. We discuss this fur-ther below, and address this technicality such that the model reproduces CASSCF for 3electron pairs.

With the truncation of T at Tn, Eq. (5.8) must be solved iteratively with O(o2n+2) oper-ations per iteration, as follows from contractions associated with the O(o2n) amplitudes. Aquantitative picture of an n-electron singlet dissociation problem often requires some am-plitudes in the nth rank cluster approximation, which means O(o2n+2) effort. For chemicalsystems undergoing 6-electron processes (e.g. the Cope rearrangement) this corresponds toO(o14) computation. This effort, while far better than exponential, is scarcely affordablefor anything beyond toy calculations at present. We must therefore follow an alternativepath that does not simply truncate the VOO-CC hierarchy by substitution level alone.

5.2.2 Definition of the model

As already mentioned in the Introduction, instead of truncating T by substitution levelalone, in the PP, PQ, PH hierarchy, amplitudes are chosen so that a certain number ofelectron pairs, p, can be treated exactly, while amplitudes that couple more than p pairsare discarded. With this simultaneous truncation by the maximum excitation level, 2p forthe p-pair model, and by the number of pairs coupled (p), the p-pair dissociation problemcan be solved with vastly lower cost than simply truncating T at T2p.

The PP model was already reviewed as motivation in the Introduction, and is a subset[63] of VOO-CCD [188]. VOO-CCD uses T2 as the cluster operator, which contains all


O(o4) double substitutions, T a1a2i1i2

. PP replaces T2 by TPP , Eq. (5.3), which contains onlyO(o) amplitudes, tj. Each amplitude belongs to a pair of two electrons, labelled as j, anddescribed by 2 orbitals, as in Eq. (5.1). Therefore all 4 indices in each double substitution

refer to only 1 particular pair: T j∗j∗

jj. This association between orbitals and pairs is made

well-defined (and informative) when the energy EPP is minimized by varying the orbitalsto find bonding and anti-bonding levels that best describe pairs of electrons. The resultingorbitals often localize in physically meaningful ways, as is mathematically required in orderto exactly describe the behavior of truly isolated electron pairs. From that PP startingpoint, we recently then defined and implemented the PQ model [86], which is exact in theactive space for isolated 4-electron systems (2 pairs). PQ is thereby a subset of VOO-CCDTQ, where amplitudes and integrals are only retained if they couple ≤ 2 different pairindexes (in contrast to PP which couples only one).

We turn now to the next level of the hierarchy, the PH model, which is designed to beexact for isolated systems of 3 electron pairs (6 electrons), as well as properly extensive.To accomplish this objective, the equations of the PH model are defined to be those ofthe valence optimized orbital coupled cluster theory truncated at hextuples (ie. VOO-CCDTQ56), where amplitudes and integrals are allowed to possess indices originating inat most three different pairs, as indicated below:

T =∑n≤6

Tn ; with Ta1,a2,...,an

i1,i2,...,ins.t.ak, ik ⊂ Pair1 × Pair2 × Pair3 (5.9)

There are an enormous variety of amplitudes that satisfy the condition of Eq. (5.9), begin-ning with every PP amplitude and every additional PQ amplitude (since they couple only1 and 2 pairs respectively).

Additionally the D, T and Q substitutions that couple 3 different pairs are included inPH. For example, if there are 4 pairs in the active space (a CASSCF (8,8) space), PH wouldinclude doubles that might be denoted as T 3α3β

1α2β , which are omitted in PQ. Their inclusion,without increasing the level of substitution, serves to reduce the truncation error associatedwith only including amplitudes with 2 distinct pair indexes in the PQ model. There is stilla remaining pair truncation error at the D, T, and Q levels of substitution. In the same 4pair example mentioned above, the double substitution, T 3α4β

1α2β , does not satisfy Eq. (5.9)and is omitted from PH. Similar considerations apply to the T and Q substitutions. Inaddition, pentuple and hextuple substitutions that couple 3 pairs are included in the PHmodel, which permits exactness for a system of 3 electron pairs. Regardless of the levelof substitution, all amplitudes retained in the PH model can be labeled with indices thatgrow cubically with the size of the system, giving O(o3) amplitudes.

PH is a much more severe approximation to VOO-CCDTQ56 than PQ is to VOO-CCDTQ. There are O(o8) quadruples in the nonlocal VOO-CCDTQ model, of which onlyO(o2) are retained in PQ – a fraction that goes as O(o−6). For hextuples in VOO-CCDTQ56and PH, these figures change to O(o12) and O(o3), meaning the fraction of amplitudes re-


tained has diminished from O(o−6) in PQ to O(o−9) in PH. This is quantified in the nextsection, but it should be clear that simultaneous truncation by maximum substitution leveland by number of pairs coupled permits levels of substitution that would otherwise bepossible only for toy systems.

Given the definition, Eq. (5.9), of the substitutions to be retained in the PH trunca-tion, the remainder of the model is specified at least implicitly by the VOO-CC machinerysummarized in the previous sub-section. There is one retained equation for each retainedamplitude, of the form of Eq. (5.5). There is one Lagrange multiplier for each retainedequation, enabling the construction of the pseudo-energy, Eq. (5.7), in terms of the re-tained equations. The pseudo-energy of the PH model must be made stationary withrespect to all energetically signficant degrees of freedom, following Eq. (5.8). We replacethe occupied-virtual orbital rotation condition with the Bruckner condition to ensure ex-actness (see below). While this prescription is clear and well-defined in outline, the PHmodel is none-the-less enormously algebraically complex – even the CCSDTQPH equationsdo not (to our knowledge) appear explicitly in the literature, let alone the VOO-CCDTQ56equations, orbital variations and all! Therefore we turn next to the challenge of implement-ing the PH model for practical calculations.

5.2.3 Single excitations, orbital-optimization and exactness

We claim exactness for 3 electron pairs but as realized by others [232], and mentionedin the PQ paper, OO-CC [233] is not generally exact without singles, except in situationswhere the odd-particle numbered amplitudes are zero because of a symmetry. Withoutthem the Schrodinger equation has not been projected against singles and expanded com-pletely over the given many-body basis, although there is very little correlation physics inthat block of e−T HeT . The error caused by this approximation is small relative to the localapproximation (tens of µEh), and usually of opposite sign, so we have avoided treating thistechnicality until now.

To satisfy the SE exactly within the active space, our model must satisfy the SE pro-jected against singles, the Bruckner [234] condition:

〈0|µ1HeTc|0〉 = 0 (5.10)

To ensure this without introducing T1, we provide a hybrid variational/Bruckner [234–236]orbital gradient as first suggested by Olsen and Kohn [232]. At each orbital iteration wecalculate the few terms of the singles residual which do not depend on T1:

F p01h02 + F p3

h1 Th02h1p01p3 +

1

2T h02h2

p4p5 V p4p5h2p01 +

1

2T h2h3

p01p4Vh02p4

h2h3 +1

4T h02h1h2

p01p4p5 Vp4p5

h1h2 → R1 (5.11)


Model Variational HybridVOD 0.000737 0.000740PQ 0.001467 0.001468PH 0.000060 0.000060

Table 5.1: Energetic effects of the hybrid gradient with several OO-CC models of water withthe 6-31g** basis. Energies reported are Model - CASSCF(8,8) in Eh. R(O-H) = 1A, and∠(HOH) = 103.1.

The active-occupied → active-virtual block of this matrix is divided by usual eigenvaluedenominator, multiplied by negative one, and inserted into the usual variational expressionfor δEc/δU

qp as a gradient descent step. δEc/δU

qp is used with the current θ to form δEc/δθ

qp

and extrapolated with DIIS alongside the amplitudes [237]. Given that the cluster model’senergy is invariant to the occupied-occupied and virtual-virtual orbital rotations (and PHis invariant to these rotations in the limit of 3 electron pairs) this recipe is now exact.

For our purposes this hybrid doesn’t have any computational cost drawbacks and is alge-braically much simpler than evaluating the Λ1 equation. However in any useful applicationthe singles omission error which exaggerates the correlation energy is much less significantthan the error induced by three-pair correlation which diminishes Ec. Quantitatively, con-sidering N2 in a 6-31G basis at 1.1A, PH is above the CASSCF (10,10) energy by 323µEhwith the variational gradient; application of the hybrid gradient increases that number to328µEh. Likewise for the case of H2O (Table 5.1) the hybrid gradient slightly increasesthe energy at a scale which isn’t meaningful for our method as it is intended. Our ap-proximate Hessian for Eq. (5.10) is of lower quality than the variational gradient, and so inthe results of this paper we always employ the latter although the code exists for the former.

5.2.4 Implementation

Coupled cluster theory becomes algebraically cumbersome as the level of substitution isincreased. The level of complexity of the working equations at a given level of substitutionis further increased (even as the scaling of the actual computational complexity is reduced!)if a sparsity pattern is imposed, such as neglect of all amplitudes and integrals couplingmore than 3 pairs in PH. Never-the-less, the theory which produces these equations iswell-known, and the rules that lead to the governing algebra have been automated byseveral groups [103, 109, 112, 238], to permit high level benchmark CC calculations. Wehave developed our own automated code generation tools for the purposes of this work,which include the additional feature essential for PH of exploiting an imposed sparsity. Ithas been discussed in detail in a separate publication [239]. For completeness, the resultingequations specifying our implementation of the PH model (i.e. the particular factorizations


chosen) are given in the supporting information.It is as important to demonstrate the tractability of the PH model as its accuracy,

so we will describe how our approximations lead to affordable scaling. At convergence∂Ec/∂Λ6 = 0, but on intermediate iterations we call this quantity the hextuples residual,R6. It is composed of 135 separate terms, which are up to quartic in the operator T . In thenon-local VOO-CCDTQ56 model, computation of this quantity dominates the CPU costin the T iterations. Consider one particular term of the hextuples residual for illustration:

T a1,b1i1,i2

(T a2,a3

j1,i3(T a4,b2

i4,i5(T a5,a6

j2,i6V b1,b2j1,j2

)))→ Ra1,a2,a3,a4,a5,a6

i1,i2,i3,i4,i5,i6(5.12)

Summation over repeated indices is implied and the parentheses indicate one choice offactorization. Assuming only a three-pair approximation on the amplitudes and integralsthe contraction of the inner-most intermediate with the next amplitude is O(o6). This iseasily seen if the contraction is re-indexed by pair labels, Pn in one of the many possibleways:

T a4,b2i4,i5

(T a5,a6

j2,i6V b1,b2j1,j2

)→ T a4,b2i4,i5

(Ia5,a6,b1,b2i6,j1

)→ T P1,P5

P6,P1(IP1,P2,P4,P5

P1,P3) (5.13)

In our code (and most others), the CC equations are represented like the middle expres-sion of Eq. (5.13), where pairwise contractions are replaced by intermediate quantities likeIa5,a6,b1,b2i6,j1

and performed in series. The 3-pair constraint on the residual dictates that theak and ik are cubic and reduces the cost of this term by several orders of magnitude.Pair labels are not used to index the summation loops. Instead the contraction algorithmskips multiplications which would produce an element indexed by more than three pairs.

In the usual CC theory the cost of the nonlinear and linear (CI-like) terms grow with thesame power, but the linear terms of this model increase as only O(o5) while the non-linearones, such as the example in Eq. (5.13) above, are order O(o6). Of course, if not just theresiduals, but also all intermediates could be indexed by three-pairs this property wouldbe restored. The resulting intermediate pair approximation seems ambitious at first blush,but would reduce the cost of the method by one more power of system size. We did notenforce the three-pair constraint on intermediates lightly because in principle the accuracynow rests on the choice of factorization, but experiments on every system we’ve examined(see representative examples in Table (5.2) ) suggest that use of the 3-pair approximationfor intermediates gives error that is insignificant relative to the accuracy of the PH model.Additionally the intermediate-pair approximation conserves the pair-exact property. It issimilar in spirit to the recursive commutator decomposition approximation introduced byYanai and Chan in their canonical transformation [204] work. In the supplementary in-formation the factorization used in this code is listed, but we emphasize that only Figure1 would be significantly altered by omission of this approximation. In other words: thiscomputational convenience does not compromise the reproducibility of the model and bothcases have an implementation.

Our pilot implementation captures the essential feature of PH: the exponent of cost.


System Local Intermediates Exact Intermediates

F2 (14,14) R=1.4A -198.914613 -198.914619

N2 (10,10) R=1.1A -109.119738 -109.119746

Cope R=1.642A(14,14) -232.848401 -232.848381Dioxirane (18,18) -188.869097 -188.869109

Table 5.2: Typical energetic effects of the intermediate-pair approximation. Frozen PP-orbitalsand 6-31g* basis employed in all cases.

To verify this a series of small diamondoid hydrocarbon molecules were chosen for tim-ing benchmarks. Their bonding grows in a three-dimensional way, but they are otherwiserandom1. The wall computation times verify that the algorithm scales more cheaply thanthe 6th power (Figure 1) of molecular size. The number of parameters in the model growsmore slowly than even the lowest order non-local VOO-CCD method, and the pre-factor issuch that storage required becomes smaller for more than 31 electron pairs (Figure 2). Theaccuracy of the CCSDTQ56 implementation was verified by comparison with 6-electronFull-CI results provided by the PSI3 [240] program package.

New algorithmic refinements are required beyond the tools already developed for thePQ model [86] to make PH tractable. If one has two tensors with quadratic numbers ofnonzero elements apiece, 4th order scaling can be realized by simply focusing on the nonzeroentries in coordinate representation. The same algorithms would scale 9th order for cubicnumbers of significant elements. In order to realize the correct scaling for PH one mustonly contract together blocks of amplitudes whose results will obey the 3-pair constraintand be able to construct those block pairs cheaply. The permutational symmetries of alltensors must also be leveraged.

Spin and pair-blocking were incorporated in such a way that spatial symmetries andfurther locality could be added as well in future work. The price paid for this generalityis that each floating point operation is accompanied by many integer manipulations whichvectorize poorly. Floating point effort of contraction is performed as a matrix-matrix orvector-double product in the BLAS package [241]. The algorithm has been described in aseparate publication [239].

Optimization of orbitals is a highly nonlinear problem and deserves more attention thanit can be given here. Our implementation of PH includes a two-step [48, 242, 243] OO-CCsolver. Formally these self-consistent orbitals are desirable, but technically it is challeng-ing to make their CPU costs outweigh their energetic benefits relative to RPP. When theorbital-optimized three-pair doubles model (3p) does not collapse these orbitals are oftenfound to be converged for PH within tolerance. Because of the significant costs of calculat-ing PH’s orbitals we observe a simple protocol for choosing orbitals, attempting increasingly

1geometries are provided in the supplement


y = 5.3134x - 2.955

R2 = 0.9861

1.1

1.6

2.1

2.6

3.1

3.6

4.1

4.6

0.75 0.85 0.95 1.05 1.15 1.25 1.35 1.45

Log ( Active Occupieds )

Log

(Wal

l Sec

ond

s)

Figure 5.1: Scaling of amplitude iteration wall-time with system size. Calculations performed onone core of an Apple XServe (Fall 08). Largest system: t-butane (24,24) pictured. Parameters ofa linear least-squares fit are inset.

y = 0.09x4 - 0.12x3 + 0.13x2 + 0.00x

y = -0.00x4 + 6.31x3 - 34.75x2 + 44.75x - 0.00

1

10

100

1000

10000

100000

1000000

10000000

5 15 25 35 45 55 65 75

Number of Active Occupieds

Num

ber

of A

mp

litud

es (T

hous

and

s)

Dim(T) CCD

Dim(T) PH

Figure 5.2: Scaling T with system size. Same systems represented by each data point as Figure1, as included in the supplementary information. Least-Squares fits by quartic polynomials areinset.

5.3. RESULTS 71

accurate orbital optimizations (PP, 2p, PQ, PH) until convergence. This compromise isproven practical in the results. The odd-numbered multiplier equations which were alsoavoided in our previous work have been included in the current implementation, along withthe contributions to the gradients which arise from the odd-hole-numbered density:

γov · ∂〈f vo 〉/∂U qp + Γovvv · ∂〈ov||vv〉/∂U q

p + Γooov · ∂〈oo||ov〉/∂U qp →

δEcδU q

p(5.14)

There are several directions where this implementation could be improved to bringit in line with the codes of related methods [244]. The model is trivially parallel up tohundreds of processors because there are literally hundreds of independent terms whichinvolve the contraction of two three-pair objects. Only cubic quantities would need to becommunicated to/from cores where 5th order CPU effort would be performed. Lastly weshould comment on the possibility of an implementation like those available for the PPand imperfect-pairing [1] models where the pair sparsity is built into loop structure andnot recomputed ”on-the-fly”. We can estimate the speedup which might result by measur-ing the amount of time this code spends in integer manipulations vs. BLAS. In a typicalprofiling run of the code roughly .6% of the process’ samples find the code in BLAS. Thework presented here is essentially a pre-requisite for the pair-indexed algorithm becausethe number of unique loops will skyrocket.

5.3 Results

The PH model was implemented in a developmental version of the QChem [192] pack-age. CASSCF calculations were performed with the aid of the GAMESS [195] packageand/or PSI3 [240]. Unless otherwise noted a model employs its own orbitals which arerestricted in all cases.

We have chosen the model systems to probe the approximations of PH, or a failure of it’sparent. Benzene addresses spatial symmetry breaking and performance. The Cope rear-rangement addresses the neglect of singles in the orbital gradient. F2 addresses active spacelocality (in the context of a simple multi-reference situation), and the Bergman reactiontests locality and rank truncation simultaneously. Our figure of merit is the non-parallelityerror vs. CASSCF (NPE). In every case we follow a continuous geometrical coordinate.We do this to ensure we follow a single orbital solution, and eliminate solution hoppingas a source of error. To follow a continuous curve with most CASSCF implementationsrequires similar care, although OO-CC is somewhat more sensitive to the guess provided.

5.3. RESULTS 72

-0.002

-0.0015

-0.001

-0.0005

0

0.0005

0.001

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Deformation Angle (Degrees)

Rela

tive

Ener

gy (H

artre

e)

PPPQ (PP Orbitals)PH (PP Orbitals)PH

Figure 5.3: D3h→D6h deformation of benzene in the 6-31G basis

5.3.1 Benzene

The resonant delocalization of π-electrons in a benzene ring, and the resulting D6h

symmetry are undoubtedly elementary to the chemistry of carbon, however local valencecorrelation models are known to favor bond alternation if the locality demanded of thewavefunction is beyond reality of the electronic structure [125, 126, 190]. A similar effectcan be seen with other popular correlation models if an insufficient basis is employed [245].Given that the 3-pair constraint treats 6 electrons quantitatively, we would expect it torepair this deficiency. Indeed we find this to be the case in both the (6,6) and (30,30)active spaces (Figure 2). Even given the orbitals of the PP model (which are severelysymmetry broken) the PH amplitude equations distort benzene by less than .1mEh. Aniteration of the CC-6 amplitude equations in the (30,30) space given 3-pair constraint andlocal intermediate approximation requires roughly 30,000 seconds of wall-time in our mostrecent implementation on one core of a typical cluster node. No spatial or spin symmetriesare exploited. Our sparse storage scheme and factorization of the CC equations requireroughly 10GB of disk for applications of this size. The deformation angle is the differencebetween consecutive C-X-C angles, where X is the center of benzene’s mass. R(X-C) =1.3A, and R(X-H) = 2.0 A.

5.3. RESULTS 73

R(Allyl-Allyl) A CASSCF (6,6) PP PQ PH1.64 -232.98024 0.00778 0.00112 0.000021.695 -232.97954 0.01074 0.00174 0.000021.75 -232.97820 0.01453 0.00259 0.000021.805 -232.97703 0.01892 0.00360 0.000011.86 -232.97635 0.02350 0.00466 0.000001.915 -232.97615 0.02794 0.00570 -0.000011.97 -232.97628 0.03208 0.00666 -0.000012.025 -232.97656 0.03589 0.00754 -0.000012.08 -232.97685 0.03941 0.00832 -0.000012.135 -232.97706 0.04268 —— -0.000012.19 -232.97713 0.04577 —— -0.00002NPE(kcal/mol) —— 23.84 4.52 0.02

Table 5.3: Energies along the D2h coordinate of the Cope rearrangement in the (6,6) space and6-31g* basis. Total energies are given for CASSCF in Eh, while deviations from CASSCF in Ehfor PP, PQ, PH.

5.3.2 Cope Rearrangment

The Cope rearrangement is a classic of organic chemistry and has been carefully exam-ined [211,212] in a quantum chemical context, especially by the group of Davidson [209,210].An interesting coordinate to examine is the inter-alyllic distance along the D2h slice of thepotential surface the reaction coordinate passes through. Two qualitatively different min-ima are found by CASSCF as this distance increases, one ”concerted” and another ofbiradicaloid character. We will examine several geometries along this curve obtained fromCASSCF optimizations with the inter-allylic distace constrained2.

The most often used CASSCF space for this problem is (6,6), the same number ofelectron pairs we have constructed PH to treat quantitatively and so we use it as anothercontrol experiment to assess the neglect of singles with the variational gradient. Table 1compares the performance of PH to its predecessor PQ. Distinguishing between the twominima separated by only 3mEh on the CASSCF surface requires resolution beyond theusual 15mEh that can be expected from PQ for a system of this size. The NPE indicatesthe difference between the maximum and minimum deviation from CASSCF. Neglect of thesingles block does not hinder PH’s ability to faithfully reproduce CASSCF for this system.

2We would like to acknowledge Troy Van Voorhis for generating these geometries during his Ph.D. andGreg Beran for a related conversation. See Supporting info.

5.3. RESULTS 74

Figure 5.4: Three representative points along the path of the Bergman reaction followed in thisstudy. From left to right, Step 0.0 (eneyne), 0.5 (near transition structure, and 1.0 (p-benzyne).

5.3.3 Bergman Reaction

The Bergman reaction is interesting for many reasons, not the least of which is its role asa fuse for a class of cytotoxic natural products with therapeutic potential. The mechanismof the reaction is well-understood theoretically and experimentally, and is known to be adifficult case [246, 247] for correlation models making it an ideal benchmark system. Theproduct: p-benzyne [215, 248], is a singlet diradical. DFT calculations can provide poorresults for such systems, even predicting a bond-stretch isomer of m-benzyne which uponfurther examination appears not to exist [249]. Adopting the eneyne and transition stategeometries of Cramer, and the benzyne geometry optimized with Spin-Flip DFT [182], wewill make a very coarse study of a reaction coordinate obtained by a quadratic interpola-tion between these geometries (Figure 2) in Cartesian coordinates3 and compare againstCASSCF(8,8) (the space employed by Mazziotti for benzynes) to assess our model on apractical problem with a strongly delocalized electronic character.

PH’s parent model PQ provides a semi-quantitative picture of the reaction coordinatewith accuracy in keeping with our previous findings, usually with correlation energies a fewmEh less than CASSCF for only a few seconds of CPU time. Immediately at Cramer’sTransition state (step 0.5) where we expect most multi-reference character, we see thatPQ over-correlates presumably because of strong non-separable correlations of more than4 particles whereas PH remains variational. These geometries likely deviate significantlyfrom the lowest energy path between endpoints, but that only adds difficulty to the problemof reproducing this slice of the CASSCF surface. Without the intermediate-pair approx-imation the cost of the calculation increases significantly, but the result does not changemuch. For meta-benzyne (step 1) the correlation energy of PH without the intermediate-pair approximation is 0.4mEh larger, and for the ene-yne the same figure is 0.2 mEh.

3The coordinates of each point are available in the supplementary information.

5.3. RESULTS 75

-229.52

-229.50

-229.48

-229.46

-229.44

-229.42

0 0.2 0.4 0.6 0.8 1

Reaction Coordinate (Arbitrary)

Ele

ctro

nic

Ene

rgy

(Eh)

CASSCF(8,8) 6-31g*

PH

PQ

Figure 5.5: Electronic energy along Bergman reaction coordinate. CASSCF(8,8) in the 6-31g*basis.

Step (arb.) CASSCF(8,8) PP PQ PH0.0 229.49761 0.02789 0.00227 0.000900.1 -229.48047 0.02530 0.00129 -0.000020.2 -229.45835 0.02487 0.00124 0.000010.3 -229.46065 0.02601 0.00224 0.000030.4 -229.45690 0.02885 0.00350 0.000120.5 -229.43533 0.03809 -0.00532 0.000920.6 -229.42281 0.03541 0.00327 0.000610.7 -229.44270 0.03237 0.00301 0.001700.8 -229.46354 0.04247 0.00842 0.001290.9 -229.47450 0.03843 0.01017 0.001181.0 -229.47767 0.03924 0.00977 0.00117NPE(kcal/mol) —– 11.0 9.7 1.1

Table 5.4: Study of Bergman Reaction in the (8,8) space 6-31g* basis. Total energies are givenfor CASSCF in Eh, while deviations from CASSCF in Eh for PP, PQ, PH


R(F-F) A CASSCF PP PQ PH1.6 -198.94906 0.09720 0.01590 0.000911.9 -198.92881 0.08053 0.01311 0.000952.2 -198.91505 0.07110 0.01049 0.00121NPE(kcal/mol) 16.37 3.40 0.19

Table 5.5: F2 dissociation in the DZ basis and (14,14) space. Total energies are given for CASSCFin Eh, while deviations from CASSCF in Eh for PP, PQ, PH.

5.3.4 F2

A pair of fluorine atoms nominally share a single bond, but the extreme reactivity of thegas and poor performance of CCSD [250,251] demonstrate significant multi-reference char-acter. The full-valence active space (14,14) is also roughly the limit of a routine CASSCFcalculation and large enough to seriously assess the energetic impact of the three-pair local-ity constraint. Table 3 quantitatively compares CASSCF, PP, PQ and PH for this problemwith the DZ [252] basis and (14,14) active space. Again PQ semi-quantitatively parallelsthe CASSCF curve whereas PH maintains accuracy throughout. Without the intermediate-pair approximation the energy does not change significantly even with this very large activespace. At 1.5A with the approximation PH yields an energy of: -198.943619Eh, and with-out it: -198.943622Eh. In the absence of symmetry are 11,778,624 determinants in the CIspace, and 11,102 amplitudes in PH.

PH can be seen as two approximations to CASSCF: rank truncation and spatial locality.This model system is directed largely at the latter approximation. We only enforce localityimplicitly through orbital optimization of the correlation energy. The rank-truncation ap-proximation is more explicit, in the sense that we know what determinants are relevant fora given bond dissociation process and we know that our model will be faithful if it containsthose determinants. These results are not predictable by construction, and interesting forthat reason.


We have presented a tractable approximation to CASSCF asymptotically containingfewer parameters than MP2 with many nice properties. The foremost of these is the sys-tematic improvement of PH over PQ which results from the well-tempered nature of thepair approximation. The pair-approximation based models (PP, PQ, PH) can now be madeinto something bigger than the sum of their parts because one may calibrate affordable cal-culations with quantitative calculations; for carbon systems this is now possible at a costwhich is tractable in many new and exciting cases of interest on a single processor. This


may be contrasted with the situation at present: if an active space larger than 16 electronsis needed there is no routine approach, but many promising candidates which may findbroader application soon [180, 205, 215, 242]. In future work dynamical correlation mustbe added to the OO-CC references in an efficient way; we have mentioned many possibleapproaches already and have some results in-hand which will immediately follow this paper.

The success of this model for first-row systems stems from the fact these atoms partici-pate in at most three bonds. Moving down the periodic table to higher nuclear charge moreelectron pairs are forced into a smaller region of space and correlate with each other. Theeffectiveness of the pair approximation in transition metal bonding is an open question;however the infrequent appearance of Lewis structures with more than triple bonds in thechemical literature offers a reason for optimism.

A comprehensive understanding of the efficiency of this model relative to other ap-proaches is also desirable, but challenging to realize. There are two obvious measureswhich can be tested easily: wall computation time and number of parameters. In a DensityMatrix Renormalization Group(DMRG) calculation the dimension of the many-body basisis a free parameter and so there is an opportunity to compare ”apples to apples”. It isnot necessarily enough to judge each active-space method without dynamical correlationbecause ultimately exact solution within the active space may prove inefficient.

The pair-approximation idea might be fruitfully combined with other formalisms. Therenormalized cluster models [116] could be combined with PH, projecting the generalizedmoments onto the cubic subset of hextuples which we know are relevant and avoiding ex-plicit construction of T6. We could apply the dynamical correlation theories [225] that wereoriginally conceived for density-matrix based approaches to our own work. Incomplete ac-tive spaces are required to address large systems, and these ideas can provide them in anaccurate and well-defined way.

78

Chapter 6

The +SD Correction

6.1 Introduction

The electronic structure of molecules can often be captured by an approximate wave-function consisting of a single determinant. This is usually the case for molecules near theirequilibrium geometries. For these situations the combination of coupled cluster methodswith techniques for tackling the basis-set problem [102] can be relied on to reproduce orpredict chemical phenomena given some polynomial amount of computational time whichis always decreasing. If more than one determinant is required to qualitatively treat aproblem, even obtaining a qualitatively correct electronic structure becomes challenging.Efforts in the quantum chemical community have largely adopted a divide-and-conquerapproach where the static and dynamic correlation problems are handled separately. TheComplete Active Space Self-Consistent Field (CASSCF) wavefunction [46] is used routinelyto solve the former, although its cost scales exponentially limiting the method to roughly16 electrons. Dynamical correlation is most often dealt with by second-order perturbationtheory [167]. Correlations near the interface can cause ”Intruder-State” problems [253],which must be pushed away by level-shifts.

This artificial separation of correlation problems is not reflected in the coupled clustertheory itself [50, 254]. It is non-perturbative, and only made unsuitable for multireferenceproblems because of the rank-truncation of the correlation problem. Even double exci-tations have significant flexibility [54], but the Schrodinger equation must be projectedagainst the significant high-rank determinants [116] to maintain pseudo-variational be-haviour. Rank truncation is motivated by Moller-Plesset perturbation theory and compu-tational convenience. For situations where perturbation theory fails [255], we can do muchbetter than rank truncation. This paper explores one such choice, based on our local,orbital-optimized coupled-cluster (OO-CC) models [86,126].

The formalism is nothing more than the standard CC theory applied everywhere inquantum chemistry (Eqns. 3-4), but with an unconventional truncation of T . Like CASSCF,


OO-CC moves the strong correlations into an active space of a few orbitals. Orbital opti-mization is also used to localize the active space, reducing the number of amplitudes weintroduce by several orders of magnitude [86] especially for the higher ranks. To treat thedynamical correlations we must allow for excitations from these configurations into theexternal space. The resulting model blurs the line between static and dynamic correlation.A strong analogy can be drawn between this model and MRCI [256–259].

The traditional coupled cluster theory has many nice properties: orbital insensitivity,size-consistency, and formal simplicity. Unfortunately no multireference version has beenso uniquely defined, although given an extended normal ordering [133] it’s formulationmay be possible. Even generalizing without sacrificing vital features required a large bodyof work [169, 260–263] which is now growing into mature implementations [201, 264–266].There are two points of departure between these ideas and the model presented here. TheMkMRCC, and its relatives are based on a wave-operator formalism [267], which presentsthe advantage that they are truly free of a 1-determinant reference [268], and several chal-lenges: redundant amplitudes, intruder states, and size-intensivity.

The second difference is the choice of an incomplete reference space. Most all de-velopmental MRCC codes have been based on complete active spaces, which make themeffectively exponentially scaling and limited in scope. Of course the wave-operator meth-ods are easy to imagine with a reduced space [269], but this isn’t often done. The ideaspresented in this paper for limiting the cost of the active-space treatment with a controlledsize-consistent approximation are transferrable to other MRCC formalisms. However in-complete active spaces are somewhat at odds with the orbital invariance feature. Indeedthe goal of a local model often becomes strong orbital variance for the sake of efficiency.Making this compromise we must be careful that the orbitals are well-defined, and well-behaved with respect to symmetry breaking. This paper exists largely to see how well wecan tackle those problems.

Shortly after the appearance of the first coupled cluster quadruples (CCSDTQ) imple-mentations [96, 97] the resulting code was adjusted to formulate a coupled cluster modelfor 2-configuration multi-reference problems [171, 172], and this idea enjoys continued ap-plication and development today under many names [270–278]. The advent of general-ranksymbolic coupled cluster codes [112, 199, 279] have made these ideas commonly available,and this work owes much to the significant literature in the area. In this paper we use theconventional name SRMRCC (Multi-reference Coupled-Cluster Theory based on a SingleReference formalism) for these models.

Given only a two-electron density matrix there are also techniques which can dynami-cally correct towards a total electronic energy [204, 226]. These have a clear strength thattheir cost is insensitive to the size of the reference space, although through extended normalordering [133] this property could be brought to any model. One of the features of this workrelative to the aforementioned models is that with minimal modification there are already(freely available) codes capable of iterating the coupled cluster equations as this paper willdescribe. Some of these are highly-optimized, well-understood, and are accompanied by

6.2. THE MODELS 80

code for their gradients and response properties. Furthermore the orbitals of OO-CC areorthogonally localized [56,122], and this sparsity can be used to construct affordable localmodels with techniques already known [57] to the quantum chemistry community. Becausethe model proposed in this paper can be described with minimal formalism, and codedwith minimal effort (beyond a general CC implementation), it offers a simple test-bed forlocal models of the total energy.

6.2 The Models

We assume the usual spin-orbital basis of orthogonal, but non-canonical spin-orbitals.These are divided into four subspaces: external occupied Ok, active occupied ik, activevirtual ak, and external virtual Ak. The active space is further partitioned into pair-quartets of spin orbitals iα, aα, jβ, bβ and all these labels are defined by the orbital-optimization of the underlying active-space cluster model. Occupied and virtual orbitalsin either space are denoted oi and vk, respectively.

These local approximations to CASSCF are uniquely defined by a number of pairsn, and any rank truncation imposed on the OO-CC amplitude. If the rank truncationis implied by the pair constraint, because n pairs can make at most 2n excitations, wecall the model the ”perfect”-”2n-tuple” by analogy with the perfect pairing model (ie:PP, PQ, PH). These models are exact for non-interacting n-pair systems by construction.If the excitation rank is separately limited we just give it a ”pair number”-”rank limit”appellation, ie: the three-pair quadruples are abbreviated ”3Q”. With certain algorithmsthe cost of 3Q scales like PH, but the former can usefully save on pre-factor. Unlike a string-based MCSCF code OO-CC is naturally extensive in all cases, and often cheap enough thatthe reliable, uniquely defined valence active space can be chosen.

The basic idea is now quite simple (but simplicity should never be regarded as a vice).We would like to dynamically correlate any determinant within a reference space, and basedon our experiences with the traditional cluster theory we say these dynamic correlations liein the space of doubles above this reference space. This suggests we allow any amplitudeof the form:

Ta1,a2,...,an,A1...Ak

i1,i2,...,in,O1...Ol; s.t.ak, ik ⊂ n-pair model, and 0 ≤ k, l ≤ 2 (6.1)

and allow this truncation to replace any concerns of rank in our cluster theory. We only dothis after the orbitals have been optimized. This truncation, which will be called Ansatz(6.1) in the remainder of the paper, includes the underlying OO-CC model, external andsemi-internal doubles, and singles over the whole space space. An n-pair model excites toat most 2n particles, and the model described above excites to 2n+2. If the frozen-core andvalence-PP active space are chosen (as we recommend) then this model doesn’t increase

6.2. THE MODELS 81

the maximum excitation rank above PQ/PH because Oi is empty. The storage costs andcomputational scaling grow with reference size and basis-set size like the usual state-specificSRMRCC [280].

We have performed experiments with another choice Ansatz (6.2) which relaxes thelocality of the active space by including any amplitude which is related to its parent by thereplacement of at most two labels (pair labels or spin-orbital labels for the external space):

T =

Ta1,a2,...,an,v1,v2i1,i2,...,im

+ Ta1,a2,...,an,v1i1,i2,...,in,o1

+ Ta1,a2,...,an

i1,i2,...,im,o1,o2s.t.ak, ik ⊂ n-pair model

Ta1,a2,...,an,v1i1,i2,...,im

+ Ta1,a2,...,an

i1,i2,...,in,o1s.t.ak, ik ⊂ n+1-pair model

Ta1,a2,...,an

i1,i2,...,ims.t.ak, ik ⊂ n+2-pair model

(6.2)

E = 〈0|e−T HeT |0〉 = 〈0|HeTc|0〉 (6.3)

〈µs|HeTc|0〉 = 0 (6.4)

Note every amplitude of Ansatz (6.1) is also an amplitude of (6.2), along with most ofthe non-local excitations although the gross storage cost scales similarly in the large-basislimit, (pairs)v2 ≈ NMO3 for PP+Ansatz 2, and (pairs)3v2 ≈ NMO5 for PH+Ansatz 2.This model is more appropriate if the reference is far from exactness, but likewise morecostly.

Given a perfect pairing active space of 2n electrons in 2n orbitals and v external virtualorbitals, in the most costly case (PHSD) and assuming a three-pair constraint on amplitudesand intermediates the storage cost of Ansatz (6.1) scales n3v2, and the CPU effort scalesn3v4 These scalings are frankly prohibitive for molecules of appreciable size, but beforewe can experiment with local dynamical approximations in the external space we mustdetermine the accuracy of this base model. The difference between these models and theSRMRCC [270] is two-fold. The active-space problem is not solved separately in a CI,and the expansion in the active-space is not complete. All amplitudes are on the samefooting (most amplitudes are not clearly static or dynamical) and every correlation is madeconsistent with the others during CC iterations. This is much like what occurs if C and Tare solved in a two-step iterative process [270] in the SRMRCC.

The simplicity of these models is pleasing. Every amplitude possesses a unique nonlinearequation it must satisfy derived from projections of the Schrodinger equation, and theseequations are well known [281]. There are no sufficiency conditions [169], or intruderstate problems [282, 283], but experience with the usual SRMRCC [280, 284] raises someconcerns. Because the coefficient of the OO-CC’s reference determinant is set to one byintermediate normalization, in a situation where its relative weight in the wavefunctiondrops to zero, convergence of the amplitude equations will become difficult. This doesn’tdiffer from the usual CC and isn’t much of a practical concern. A denominator shift

6.3. RESULTS 82

gradually relaxed during iterations is usually all which is required. In a similar vein,only the formal determinant experiences the highest rank of excitation and so when thedominant configuration changes abruptly the correlation energy may not behave smoothly.In the literature of SRMRCC this is called a Fermi vacuum or orbital invariance problem[279, 280]. Luckily the CC-theory is strikingly insensitive to choice of reference becauseof the exponential single excitations, and our model includes these naturally between allspaces. Moreover, orbital-invariance was already sacrificed to introduce locality into theunderlying OO-CC model, a sacrifice which must be made to escape exponential scaling.We emphasize that our goal for this work was to demonstrate the strength of the localOO-CC as a reference in a conceptually straightforwards, but quantitative model. Like anyformal drawbacks, the final measure should be the results.

6.3 Results

For the purposes of this paper ansatz (6.1) will be called ”+SD”, and that will be at-tached to the name of the OO-CC model underneath (ie: PPSD, PQSD, PHSD, etc...).Applications of ansatz (6.2) will be specifically noted. Our figure of merit will be the non-parallelity error (NPE), the difference between the maximum and minimum error relativeto the exact result. In future work the external space should also be made local. Untilthen larger active spaces will imply a more local model and necessarily smaller correlationenergies. Still if the pair approximations are good (and they usually are) the NPE shouldremain good and this will be shown below.

We employ some common benchmark systems to evaluate the performance of our mod-els. The double dissociation of H2O demonstrates the compactness and chemical accuracyafforded by this truncation. F2 largely probes the impact of active-space locality. BeH2

is troublesome from an orbital invariance standpoint, but the PQ model is non-local andrank-complete for the reference problem. H8 provides an example where orbital invariance,symmetry and locality are all challenging at once.

The ”+SD” models have been implemented in a developmental version of the QChem[192] package. Exact results were furnished by the PSI3 [240] program package. We alsocompare against the previously available dynamical correlation correction to OO-CC [230]developed by our group. The valence PP active space is generally assumed, as is the frozen-core approximation. The basis is generally DZ [285] as obtained from EMSL’s Basis SetExchange [286,287].

6.3.1 H2O

We examined the simultaneous dissociation of water with PQSD and the (8,8) activespace, the errors relative to complete solution in the DZ basis with the frozen-core approx-

6.3. RESULTS 83

-0.0020

0.0000

0.0020

0.0040

0.0060

0.0080

0.0100

0.0120

0.0140

0.0160

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Err

or

(Eh)

R(O-H) (Angstrom)

UCCSD Error

PP+SD Error - .0606

PQ+SD Error

PH+SD Error

Figure 6.1: Simultaneous dissociation of H2O in the DZ basis. Orbitals are those of the GVB-RCCmodel [1].

imation are pictured in Figure 1. In the absence of symmetry the complete expansion is1,656,369 determinants and there are 2,745 amplitudes in CCD. PQSD is a significant trun-cation of the correlation problem with only 2,984 amplitudes, but maintains a very modestnon-parallelity error (NPE) of 1.07 kcal/mol, re-enforcing the efficiency of the pair-basedreference. Even with rather crude PP orbitals PH+SD maintains an NPE of 0.19 kcal/mol.

6.3.2 F2

Electrons are loosely paired between Fluorine atoms in a multi-referenced bond thatis poorly described by low-rank single-reference cluster theory [288, 289]. We also includethe best unrestricted curve which can be coaxed from B3LYP [290] for any newcomers tothis area. The valence perfect-pairing space, (14,14), is also roughly the limit of completediagonalization, and so this one of the more stringent tests of the locality approximationwe can produce. We’ve previously demonstrated that the PH model provides chemicalaccuracy for the valence CASSCF correlation problem. In the same study we saw thattwo-pair locality can only provide an NPE to CASSCF of roughly 4 kcal/mol.

The truncation introduced in this paper does not relax active-space locality, and so weexpect the errors relative to CASSCF to translate almost quantitatively into error relativeto the total correlation energy (if we have captured the bulk of the dynamical correlations).Unfortunately we are unable to provide a FCI curve, but for the purposes of this work thecomplete CCSDTQ model should provide near-exact results and the errors for this exampleare reported relative to this benchmark. The core orbitals are frozen in all cases. In the

6.3. RESULTS 84

-0.025

-0.015

-0.005

0.005

0.015

0.025

0.035

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

R(F - F) Angstrom

Err

or

(Eh)

PPSD - 0.1032

PQSD

PHSD

PP(2)

B3LYP + 0.5564

CCSD

Figure 6.2: Dissociation of F2 in the DZ basis, Errors relative to CCSDTQPH (frozen-core).PHSD employs the intermediate-pair approximation.

nonlocal model there are 4,597 doubles, 112,236 triples and 1,697,198 quadruples. In the3-pair model those figures are reduced by factors of roughly 1

2and 1

10respectively.

The PH paper also introduced an intermediate-pair approximation which offers a palpa-ble reduction in computational complexity of a three-pair model. This approximation onlytakes effect when the number of active pairs significantly exceeds the locality constraint(as it does in this example). The same idea can be applied to the ”SD” model without anymodification. The results of Figure 2 are obtained in the presence of that approximation.If this approximation did not carry to the ”+SD” models accurately, other avenues wouldhave to be pursued to relax the cost of this method. One direction would be including onlyrank density-matrix information from the reference as is done in Canonical Transforma-tion [227].

6.3.3 BeH2

Insertion of Beryllium into a hydrogen molecule is a classic multi-reference problem be-cause of an avoided crossing in the intermediate region. Of the examples recently considered

6.3. RESULTS 85

-0.005

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0 0.5 1 1.5 2 2.5 3 3.5 4 E

rro

r (E

h)

R(Be - H2 Center)

PQSD Error

MkCCSD Error

CCSD Error

Figure 6.3: Insertion of Be into H2. PQSD is built on the (4,4) active space. Orbitals are thoseof PQ.

by Hanrath [280] with several MRCC approaches this system posed the most difficultly forSRMRCC. The dominant reference changes symmetry between endpoints with the highest-occupied molecular orbital shifting(3a1 → 1b2). We employ the geometry and basis set ofEvangelista [265]1 from whom we also borrow benchmark results. At either endpoint theorbitals are optimized for PQ from a guess of GVB-RCC [68]. All internal data points usean orbital guess provided by the previous geometry.

In Figure X the errors relative to FCI are shown for MkCCSD(2,2), PQSD(4,4) andCCSD. The active spaces employed in MkCCSD (2,2) and PQSD (4,4) are not the same.We stress that MkCCSD possesses strong orbital invariance properties which this modeldoes not, and the fact that they perform similarly in different active spaces is a reflection ofthat strength in MkCCSD. This graph suggests that orbital-invariance isn’t a fatal concernfor these PQSD. With larger spaces the locality we exploit in PQ or PH is a more significanterror.

1We would like to thank the authors of this important paper for providing very clear and complete datathroughout.

6.3. RESULTS 86

α/π FCI PQSD Error (Eh)0.0 -2.07753 0.003210.1 -2.17098 0.004490.2 -2.21589 0.004380.3 -2.23302 0.004330.4 -2.23958 0.004340.5 -2.24131 0.00434NPE (kcal/mol) 0.80109

Table 6.1: The H4 model system. Zero is a 2-Bohr square of hydrogen atoms, and .5 is the linearconfiguration. The orbitals are those of PQ. The reference space is (4,4) and basis DZP.

6.3.4 H4

Four hydrogen atoms are arranged in a 2 Bohr square, and the opposite sides of thatsquare are folded downwards by an angle determined by parameter, α. As a square (α =0) the electronic structure is dominated by two equally important determinants which areseparate as the structure is made linear (α = 1

2). This model was introduced by Jankowski

and Paldus, and has been previously examined by several authors [201,280,291,292].Table 1 summarizes the error of PQSD relative to FCI in the DZP basis. Given that

this is a 4 electron system PQ is not a local approximation and the anstaze closely resem-bles SRMRCC. The orbitals are orbital-optimized doubles-quadruples rather than CASSCF(2,2), and there is no CC/CI separation. There are far fewer amplitudes in PQSD (501)than the complete CCSDTQ model(6529). The performance of PQSD is satisfactory, butpredictable. We include this example mostly for the sake of completeness.

6.3.5 H8

In this standard MRCC test system [201, 291–295] eight hydrogen atoms are arrangedin a 2 Bohr octagon. Two opposite faces are pulled away from each other by a displacementα (Figure 4). As in the previous examples two determinants differing by the replacement ofan orbital (b1g → ag) exchange dominance in the wavefunction becoming quasi-degeneratewhen the molecule is symmetric. MkCCSD built on CAS(2,2) has an NPE of 2.019 mEhfor this system [201], a PP(2) calculation with the (2,2) space has an NPE of 18 kcal/mol.Table 2 examines the the errors of PPSD(2,2) and PHSD(8,8) (with intermediate pair ap-proximation) relative to complete solution in the DZ basis. The orbitals are optimized forthe PH reference. In the (2,2) space there are no symmetry breaking problems for PPSD,and we obtain a curve of quality comparable to MkCCSD. The most inexpensive OO-CCmodels (PP, IP [1]) over-localize orbitals, and under-correlate weak bonds at the expense


Figure 6.4: Graphical depiction of H8 model for α = -0.4, 0.4 Bohr

of stronger bonds. Aside from the same multi-reference problems addressed elsewhere inthis paper, this symmetry breaking presents a real challenge for a 3-pair local model of thevalence electronic structure in H8.

For the compressed geometries the three-pair approximation and PHSD(8,8) fares well,and the orbitals can be converged easily. When α > 0 the 3-pair orbitals become sub-stantially more difficult to converge than the non-local doubles indicating that the localapproximation is being forced on a somewhat non-local problem. The errors shown in Ta-ble (6.2) for α > 0 reflect errors of 3p against CASSCF much more than they reflect anydeficiency of ”+SD” itself. This motivated us to develop the less-local Ansatz 2.

However Ansatz 2 restores quantitative accuracy, even when built on a two-pair refer-ence as shown in Figure (6.5).


The pair-approximation based models have been developed with computational costand practicality as the guiding principle. The assumption that an accurate total energymodel could be made from PQ or PH given positive comparison to CASSCF has beenjustified in this paper. Because they make the valence active space affordable, and becausethe OO-CC orbitals are well-defined Fermi-vaccuum invariance isn’t the main challenge forthis scheme. The symmetry breaking inherited from GVB, and largely ameliorated with a3-pair constraint is the dominant source of error.

We do not claim that this work is the grand-unified, black-box correlation model, butthe results indicate that the ideas introduced are useful. Combinations of OO-CC withother ideas like MkCC [169], the anti-hermitian contracted Schrodinger equation [226] or


α(Bohr) FCI (Eh) PPSD (2,2) Error 3QSD (8,8) Error-0.4 -4.35706 0.00085 -0.00141-0.3 -4.35063 0.00101 -0.00178-0.2 -4.34176 0.00134 -0.00222-0.15 -4.33701 0.00167 -0.00238-0.1 -4.33266 0.00223 -0.00232-0.05 -4.32954 0.00314 -0.001690.0 -4.32872 0.00425 -0.000040.05 -4.33074 0.00304 0.009290.1 -4.33509 0.00209 0.00526

NPE kcal/mol 2.13 7.32

Table 6.2: Automerization of H8. The basis is DZ. MRPH employs the intermediate pairapproximation

-0.001!

0.001!

0.003!

0.005!

0.007!

0.009!

0.011!

-0.5! -0.3! -0.1! 0.1! 0.3!

Mod

el -

Exac

t (H

artr

ee)!

Alpha (Bohr)!

UCCSD(T)!Ansatz 2 + 3Q!Ansatz 2 + PQ !Ansatz 2 + PP !

Figure 6.5: Automerization of H8. The reference space is (8,8) and basis is DZ.


Canonical Transformation [227] would also be interesting.Given the attractive simplicity of this work, it is interesting to ask how this dynamical

correction might be made as efficient as it’s static correlation counterpart. We are workingoff this footing to produce a entirely local correlation model. Along similar lines somerecent studies [296,297] made local correlation models from the renormalized cluster theory.Another fascinating direction would replace the Gaussian orbitals of the external spacewith explicitly correlated geminals. Given sustained growth of computational resources,systematically-improvable approximations like these whose cost grows with some reasonablepolynomial are poised to make a large impact.

90

Chapter 7

A Novel Range Separation ofExchange

7.1 Introduction

Despite rough beginnings [25], the local density approximation (LDA) has been devel-oped through decades of work on the Kohn-Sham (KS) [80] construction into one of themost successful approximations in quantum chemistry and solid state physics. Within thisframework, the LDA exchange correlation functional is combined by an adiabatic connec-tion with a non-interacting wavefunction so that an approximate kinetic energy may beextracted and there is no need to develop accurate functionals for the kinetic energy [298]which have proven elusive. Along these lines, Becke [290] realized that the accuracy ofKohn-Sham energy functionals could be improved by the admixture of ”exact” exchangecoming from the explicit exchange energy of the fictitious Kohn-Sham wavefunction. Theresulting hybrid density functionals have been the most commonly applied model chem-istry for many years [299] because they have been found to be remarkably accurate withcomputational costs virtually equivalent to those of the Hartree-Fock (HF) method.

One of the few remaining substantial defects of the Kohn-Sham construction whichhas attracted theoretical effort is the so-called self-interaction problem [300–306], and it isdirectly related to the treatment of the exchange energy [83]. In the HF energy expres-sion the Coulomb repulsion of a 1-electron function with itself is cancelled exactly by thecorresponding exchange integral. In the KS construction with a pure, local functional theCoulomb energy is non-local, but the exchange energy is not. Considering the 1-particlefunctions provided by the KS wavefunction we might say that the electron repels itself ifthe particle is spread over space because the antisymmetric complement of the Coulombinteraction, non-local exchange, is missing. At equilibrium geometries the effect on pre-dicted ground state energies is not severe, but this defect means that dissociation problemsmay lead to fragments which only possess a fractional number of electrons [307,308], or re-


sponse properties which reflect serious artifacts if charge is significantly redistributed [309].If globally a fraction of the exchange energy of the KS determinant is mixed with the DFTexchange energy these artifacts are partially remediated. To a stranger unfamiliar with thehistory of hybrid DFT’s development the situation must seem confusing, because it is notobvious why any mixture of ”exact” exchange with (semi-)local exchange is advantageous.The answer is that the accuracy of most DFT functionals lies in a cancellation of errors be-tween exchange and correlation functionals. Both are compensating for the single-referencenature of the fictitious KS determinant [310,311]. From another angle, one might say thatthese two non-local objects [312] are best considered together because what results is morelocal.

A way to preserve the local cancellation of errors yet recover correct exchange at longrange has emerged in the form of range separated hybrid functionals. The idea which goesback to the pioneering work of Gill and Savin [214, 313, 314], is to divide 1/r by multiply-ing it with a function which varies between 0 and 1, such that both this function and itscomplement are integrable. The greatest fraction of work in this very active area [315–320]has employed the standard error function to achieve this separation:

1/r = erf(ωr)/r + erfc(ωr)/r (7.1)

The LDA exchange functional corresponding to erfc(ωr)/r, and integral kernel for the ex-act exchange over erf(ωr)/r can then be derived so that locally exchange is provided bythe LDA and at a distance exchange is provided by the KS wavefunction. The positionwhere the transition is smoothly made between the two treatments is determined by theadjustable parameter 1/ω. The choice of the error function as a Coulomb attenuator is bothpractical (for most implementations one must be able to perform the integral of Gaussiansover the function analytically [321]) and arbitrary because the error function is just oneof many which possess this property. In some recent studies promising results have beenattributed to more flexible range separation [322].

One can extend the idea of mixing ab-initio and DFT strengths further by imaginingrange separation of the correlation part of the functional as well. In this scheme (which wewill not pursue in this work beyond mention) the ab-initio method is made responsible forstatic and long range dispersion effects while the LDA correlation functional is adjustedfor the modified coulomb interaction to avoid double counting. Savin and coworkers haveexperimented with the choice of another attenuated Coulomb interaction for these purposes[323] , a linear combination of an error function and a Gaussian which offers a sharperseparation:

vee,erfgau = erf(ωr)/r − (2µ)/√π ∗ e−(1/3)µ2∗r2 (7.2)

More recently this erf-gau LDA functional was combined with a standard GGA in anattempt to surpass the accuracy of exchange-hybrid functionals based on erf [324] with

7.2. THE TERF-ATTENUATED LDA 92

a GGA correlation treatment. Another 1-parameter attenuated Coulomb interaction, theYukawa potential, has also been the subject of recent investigations [325,326].

vee,yukawa = exp(−γr)/r (7.3)

Improved performance of the resulting functional was attributed to an increased fractionof short-range exchange [327].

Our group has recently published an analytical integral over a more general sort ofCoulomb attenuator which allows for separate control of where and how rapidly the shiftis made between parts of the Coulomb interaction [328]. It allows continuous variation ofsharpness between the limits of erf and the Heavyside function. The function is a linearcombination of two error functions (although note that the erf formulas aren’t sufficient todescribe it) and so we have adopted the name ”terf”:

terfr0,ω(r) = (1/2)(erf(ωr − r0) + erf(ωr + r0)) (7.4)

Investigations into the performance of range-separated hybrids [329] have found that exist-ing separations cannot simultaneously describe ground-state and excited state propertieswith a single choice of ω. Along these lines a common area of intersection has been locatedamongst many optimized attenuators at roughly .8 Bohr [330] in the (r, Voptimal(r)) plane. The physical implication is that cancellation of GGA-exchange and GGA-correlationerrors in this region is balanced with the error induced by semi-local exchange, and ourattenuator should be shaped similarly in this region for thermochemical accuracy. Yet torepair self-interaction error the attenuator should reach its asymptote as rapidly as possibleonce we leave this region. The terf functional form can pass through this point while stillreaching its asymptote more rapidly than erf, and so there is reason to hope that terf maybe useful in this respect. Another nice feature of this choice of separation is that it reducesto the erf attenuator if the r0 parameter is chosen to be zero. The attenuator is plotted forseveral choices of parameters in Figure 1.

7.2 The terf-attenuated LDA

The exchange energy of a many-fermion system, charge balanced by a structurelesspositive background is our starting point. This matrix element is [331]. (where θ denotesthe Heaviside function)

Ex =−k3

f

12π4

∫ ∞0

q2vee(q)(1−3

2x+

1

2x3)θ(1− x)dq ; where: kf = (3π2n)1/3, x = q/(2kf )

(7.5)

7.2. THE TERF-ATTENUATED LDA 93

0 1 2 3 4 5r Ha.u.L

0.2

0.4

0.6

0.8

1.0

AttenuatorHrL

2, 0.98, 0.1382, 1.329, 0.3161.4, 1.345, 0.2481.016, 1.2, 0.15771, 1.48, 0.23950.4, 0, 00.3, 0, 0.1577

Ω, r0, cx

Figure 7.1: Various attenuators plotted for comparison. The first two are equivalent to Erf (ω =.3, .4).

So we must obtain the Fourier transform of terfcr0,ω(r)/r.

Fterfcr0,ω(r)/r = vee(q) =4π(1− e−

q2

4ω2 cos( qr0ω

))

q2(7.6)

The integration of this function is algebraically quite tedious, but can be done. Unfor-tunately the complex error function enters. Note that for z ∈ C, erf(z∗) = erf(z)∗ andfor z ∈ R, erfi(z) ∈ R. We report the exchange energy per particle of the Fermion gasexperiencing this interaction, εxc which may be readily implemented in any KS-DFT code.

Ex =

∫n(R)εxc(n(R))dR (7.7)

7.3. APPLICATION TO RANGE SEPARATED HYBRIDS 94

The spinless kf can be easily replaced to obtain the spin-density functional.

εω,r0x (n) =ω4

192A4π3

(

8Ae−r20√π

(Ar0(3 + A2(4r2

0 − 6))(erfi[r0] + Im(erf(1

2A− ir0))) + Re(erf(

1

2A− ir0)

))+(−(3 + 24A2 + 32A4(r2

0 − 1)) + e−1

4A2 16A2((1 + 2A2(r20 − 1))cos(

r0

A) + Ar0sin(

r0

A)))

where: A =ω

2kf(7.8)

One may easily verify that this expression matches the known erf expressions for εx asr0 → 0 [313, 332]. For large values of A (>1), a series expansion in powers of 1/A is em-ployed up to 10th order in our implementation for purposes of numerical stability.

7.3 Application to Range Separated Hybrids

Without semi-local gradient information the thermochemistry of this functional wouldbe undoubtedly poor and it would be difficult to determine if terf could improve function-als in use today. There are several recipes for combining this LDA exchange functionalwith a GGA enhancement factor ranging in degrees of technical difficulty and empiricism.Ideally the GGA factor will depend on the attenuating parameters [333], but recent resultshave shown that superior accuracy [83] can be obtained even if this is only done implicitlythrough optimized parameters of the GGA. At the end of the day the choice of ω param-eter is quite empirical, as will be r0, even if we introduce them for physical reasons. Thefinal measure of a range-separated hybrid is optimization over a large training set, andevaluation over an independent test set, roughly a year of computer effort. We seek somejustification for such an effort and so we combine the terfc-LDA exchange energy with theGGA exchange enhancement factor and correlation functionals of ωB97X [83] and runsome basic tests to establish whether the resulting functional shows promise. To be pre-cise, the resulting functional is obtained directly from ωB97X by replacing the F (aσ) ofequation (7) in that paper with the corresponding terfc-LDA F (aσ) obtained from equation(8) above. To begin from a functional as close as possible to the parent (see the previouspaper to clarify the notation), we also incorporate a variable fraction of short-range HFexchange in such a way that the UEG limit is respected (Eqns 9, 10).

ESR−DFAx =

∑σ

∫eterfc−LSDAxσ (ρσ)gωB97X

xσ (s2σ)dr (7.9)


-5.02

-5

-4.98

-4.96

-4.94

-4.92

-4.9

-4.88 0 1 2 3 4 5 6 7

Ener

gy (E

h)

R(He-He) Angstrom

Terf (1.016) wB97X Correct Asyptote

Figure 7.2: Dissociation of He+2

-1054.56

-1054.54

-1054.52

-1054.50

-1054.48

-1054.46

-1054.44 2 3 4 5 6 7

Ener

gy (E

h)

R(Ar-Ar) Angstrom

Terf (1.016) wB97X Correct Asymptote (Terf) Correct Asymptote (Erf)

Figure 7.3: Dissociation of Ar+2

Exc = ELR−HFx + cxE

SR−HFx + (1− cx)ESR−DFA

x + EωB97Xc (7.10)

Aside from the many parameters associated with the GGA we must choose reason-able guesses of r0, ω, cx. The physically motivated guess is to reach the asymptote asrapidly as possible while still overlapping significantly with the established attenuators in


ω (a.u.) r0 (a.u.) cx MAE He+2 Error Ne+

2 Error Ar+2 Error Kr+

2 ErrorBLYP† - - - -87.55 -80.37 -48.51 -49.0

0.3∗ 0 0.1577 2.09 -38.9 -34.4 -14.0 -11.20.4∗∗ 0 0 2.53 -35.1 -33.3 -9.7 -7.7

1 1.48 0.2395 5.71 -31.7 -29.3 -6.7 -4.21.016 1.2 0.1577 4.02 -24.9 -24.7 -3.0 -1.51.4 1.345 0.248 5.77 -26.6 -26.2 -2.9 -1.22 1.329 0.316 9.16 -23.2 -21.9 -0.4 0.02 0.98 0.138 7.38 -13.0 -16.2 -.03 0.0

Table 7.1: Mean absolute error (kcal/mol) of G2 set atomization energies and errors of dimercation asymptotes for various functionals. *(ωB97X), **(ωB97) †(pure Becke 88 exchange [2] andLYP [3] correlation, errors in this row are upper bounds.)

the previously mentioned critical region [330]. An initial choice of parameters r0, ω, cx =1.2, 1.016, cωB97X

x was made by this physically motivated criterion, and the usefulness ofthe resulting functional was assessed on noble gas dimer-cation dissociation (Figures 3,4).Even with only very conservative changes made to the functional form of the attenuator,terf was able to significantly increase the accuracy (relative to its predecessor ωB97X) ofthe dissociation asymptote associated with the self-interaction problem (SIE) (Figure 4).In the case of He+

2 the valence density lies so close to the neighboring atom that it seemschallenging to reach a compromise between thermochemistry and exact exchange within atransferable range-separated exchange functional but the results for Ar+

2 are encouraging.The solid thermochemical performance of the parent functional seemed more-or-less con-served, and so we investigated a little further. We make the approximation that the GGAparameters are unchanged between Erf (ωB97X) and Terf attenuated coulomb interactions.Undoubtedly this should be improved upon in future work and the literature already de-scribes many ways this may be done.

A few more sets of attenuator parameters were obtained by maximizing the least-squared overlap of a terf attenuator with those of ωB97 and ωB97X varying r0, cx fora given ω. Noble gas dissociation curves and atomization energies were calculated for thestandard G2 [334] thermochemical test set in the 6-311++G(3df,3pd) basis with a satu-rated quadrature grid. The purpose was not to search for parameters which would surpassωB97X, because gradient corrections are vital to thermochemical accuracy, but rather todocument the balance between thermochemistry and correction of the self-interaction error(Table 1). As expected the results were quite sensitive to the steepness of the attenuatorand the amount of middle-range exchange, but note that the further this attenuator de-parts from Erf the more severe becomes the GGA approximation. An accurate asymptotecannot be obtained by simply increasing the amount of exact exchange in the attenuator


(because eventually the correlation part of the problem is disturbed), but by maximiz-ing overlap with the established ones in the critical region we do obtain reduction of SIEwith increasing ”exact” exchange. At the moment where the singly-occupied MO’s bondis breaking if this MO’s density around atom 2 is far enough from the bulk of this MO’sdensity around atom 1 so that it experiences Hartree exchange the density will localize onan atom (as it should physically). Smaller atoms cause greater difficulty in general de-pending specifically on shell structure. The results suggest that if the GGA enhancementfactor of the functional were polished it would possess thermochemistry much like ωB97Xwith a significantly larger amount of exact exchange, and even in its current incarnation isaccurate enough to be used in lieu of others for problems where ”exact” exchange mightbe important.


Owing largely to the popularity of hybrid functionals range separation of exchangehas become an intense area of research, and more flexible range separation may prove de-sirable [311]. Indeed this has already been done with the erf-gau type attenuator [324],although in this case this was done at the expense of abandoning a physically motivatedchoice of a parameter. An analytical formula [328] is available for the exact exchangeenergy with the terf attenuation, and this expression has already been efficiently imple-mented in the publicly available release of the Q-Chem package [192]. This paper providesthe other building block, the short-range LDA exchange energy and a proof-of-conceptGGA functional. Preliminary results with unoptimized parameters indicate that the newfunctional may be a useful improvement and development should continue in the area ofmore general exchange attenuators. Further improvement over the functional developedhere might realized through complete reoptimization [83]. Alternatively one could derivethe corresponding PBE type GGA-functional [333,335,336]. In either case, the path is clearand only limited by one’s curiosity. It will be especially interesting to see if the flexibilityof the new attenuator can simultaneously describe ground state electronic structure andexcited states. Special attention should be paid to the size of the chromophore relative tothe scale of the attenuator, and the distance over which electron density is redistributed.This direction is currently being pursued in our group.

98

Chapter 8

Conclusions & Outlook

8.1 Summary

In this work we have presented a convergent hierarchy of approximations to the CASSCFmethod for strong valence correlations and the extension of those methods to a models ofthe total electronic energy. The cost of these methods grows with accuracy and systemsize in a tractable, polynomial fashion and systems much larger than those which can betackled with CASSCF can be addressed already with the implementations developed here.The most accurate choices in this menu afford chemical accuracy for the test cases we haveexamined. We have also presented a density functional with reduced self-interaction error.

If a chemist is interested in a problem of multiple broken bonds or otherwise uncertainabout how to assign a certain number of formal bonds a typical approach with existingtechnology may have been difficult. One of these local, high-rank cluster approximationscould provide significant accuracy and insight. Because they are well-defined for a givenchemical situation it is obvious how one should proceed. The user can begin with the mostaffordable option, choose the well-defined and reliable valence active space, and proceeduntil convergence or exhaustion of computational resources.

As said on the first page of this thesis, the purpose of this work, and the substance of theinsights are to express the wave-function with less and less complexity. In that respect wehave indeed made a little progress. With a 5th order number of variables we can offer a veryaccurate and reliable model for chemistry. A companion problem is to efficiently calculatethat model on computers which we have in hand. We have demonstrated that it is possibleto do so, but the computer science which would make it as efficient as implementations ofDFT is an open question.

8.2. FUTURE RESEARCH DIRECTIONS 99

8.2 Future Research Directions

The models we have presented are so complex that they could not be perfected inthe duration of a single Ph.D. and there are several points of entry where they could beimproved significantly and rapidly. The directions of improvement are both physical andalgorithmic. In both cases new and interesting systems could be calculated with new levelsof accuracy and it seems likely that these directions will be pursued soon.

Excited States

One clear physical shortcoming is that these models have been developed with onlythe electronic ground state in mind, but excited states need even more strong correlationand are poorly understood. A starting point would be to exploit the CC underpinningsof these models and take an Equation-of-Motion approach, diagonalizing the similarity-transformed Hamiltonian for excited states. Such an approach would have limited accuracybecause the localized orbitals (which have been optimized to fit the ground state) are notequally appropriate for the target excited states. In order to achieve an even-temperedtreatment, the response of the orbitals to excitation must be included. The formalism forthese responses could follow the example of Linear-Response MCSCF [337].

Efficient Dynamical Correlation

+SD was presented as an iterative correction to an active space correlation model, how-ever it’s likely that a perturbative correction would provide intermediate accuracy at asignificantly smaller cost, and with more easily optimized code. Just as MP2 arises natu-rally as the first iteration of CCSD, the +SD correction may be calculated perturbatively asthe first iteration of it’s iterative counterpart. In most applications that correlation energylies between the active-space reference and the converged energy. The success of a perturba-tive model relies on a cancellation of errors. Before a specialized code is developed for sucha model, this cancellation of errors should be carefully evaluated over several model systems.

Density Functionals

Range-separated density functionals have emerged as a near systematic improvementover global hybrids like B3LYP because they repair a physical deficiency of the base func-tional, but the more general terf -separated version would likely offer even better perfor-mance. To develop such a hybrid with the code currently available one should begin withthe GGA optimized for ωB97X, and develop a test set which includes several difficult ex-change problems then perform a simultaneous optimization of both GGA and attenuatorparameters. Isomerizations of branched, saturated alkanes are emerging [338] as one such

8.2. FUTURE RESEARCH DIRECTIONS 100

class of difficult problems, which might otherwise seem perfectly well-suited for DFT.

Algorithmic Improvments

The code itself limits the scope of possible applications and so I’ll outline the placeswhere it can be most rapidly improved. It is difficult to appreciate how significantly theperformance of a program can vary based on memory access patterns. To deliver codeswhich scale properly with system size we introduce significant logical overhead to avoidcalculating the parts of the correlation problem we truncate. In the codes developed forthis work that logical overhead always takes the shape of just a few operations: lexicalsorting of an array of integers, ripping stripes of integers from arrays, and collecting thoseintegers into tagged blocks. Although some attention has been paid to each operation, noneof the relevant routines even approach the optimal operation count of a modern processorbecause a significant amount of time is spent waiting for information from distant regionsof memory. Careful computer science on each operation would recover significant perfor-mance boosts.

A spin-orbital formalism has been employed, although the target systems usually con-tain an equal number of α and β electrons. The symmetry between these two sorts ofspin-orbitals is not exploited in the code (ie: each αα contribution is calculated separatelyfrom ββ even though these are the same). In general each amplitude has an identical spin-flip partner which is redundantly calculated, but should be omitted. If it were, storagecosts would immediately be halved.

The electron correlation models we have presented are composed of literally hundredsof nonlinear terms, however there is a significant degree of repetitious effort which can beavoided by proper factorization. For example each term in the coupled-cluster multiplierresidual originates in a term of the amplitude equations, and if that amplitude residualterm is stored, we can avoid repeating that calculation, algebraically:

V T n = V T n−1T → ΛV T n−1 (8.1)

Save and reuse: V T n−1 (8.2)

This algorithmic improvement would reduce the cost of the gradient by roughly a factor oftwo.

Self consistent orbitals require repeated integral transformation which costs 5th order,the same exponent governing the cost of the correlation calculation itself. However by em-ploying a tensor decomposition approximation like the resolution of the identity [127, 222]the pre-factor on the transformation operation can be reduced by a factor of more than ahundred. Algebraically implementing this change in the code would be accomplished by re-placing the integral in each term with the contraction of two tensors over an auxiliary basis.

101

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